CE Board Exam Randomizer

Question 1

Circle

A circle is the locus of all points that are a fixed distance (radius) from a fixed point (center).

Standard form:

$$ (x - h)^2 + (y - k)^2 = r^2 $$

(h,k) is the center of the circle

Answer

Equation of the Curve Connecting Centers of Circles Tangent to the Fixed Circle and the x-axis – Diagram

Observation: The equation is quadratic in $x$ but only first degree in $y$. A curve of the form $$Ax^2 + Ey + F = 0$$ is a parabola. Hence, the locus is a parabola.

Answer

Step 1: Subtract the equations of the two circles

C₁:  x² + y² + 2x + 4y − 3 = 0  
C₂:  x² + y² − 8x − 6y + 7 = 0
--------------------------------
       10x + 10y − 10 = 0

Simplifying:

$\boxed{x + y = 1}$

This line is the common tangent at the point of contact.

When two circles are tangent internally or externally, their point of tangency lies on a line that is perpendicular to the line joining the centers. The key idea: The difference of the equations of two circles gives the radical axis—the locus of points having equal power with respect to both circles.

If two circles intersect at two points, the radical axis is their common chord. But if the circles touch at exactly one point (tangent), the radical axis collapses to the common tangent at the point of contact.

Therefore, since subtracting C₂ from C₁ yields a linear equation, it must represent the common tangent.

Answer

Finding the y-coordinate of the Point of Tangency – Diagram

Step 1: Determine the radius using the tangency to the y-axis

The distance from the center $(2,3)$ to the y-axis $(x = 0)$ is: $$ r = |2| = 2 $$

Step 2: Use the right triangle formed by the center and point of tangency

Let the point of tangency be $P(3, y)$. The horizontal distance from the center to $P$ is: $$ 3 - 2 = 1 $$

Let the vertical distance be $h$. Using the Pythagorean relationship: $$ h^2 + 1^2 = r^2 $$ $$ h^2 + 1 = 4 $$ $$ h^2 = 3 $$ $$ h = \sqrt{3} $$

Step 3: Compute the y-coordinate

$$ y = 3 + h $$ $$ y = 3 + \sqrt{3} $$

Final Answer:

$$ \boxed{y = 3 + \sqrt{3}} $$

Answer

Substitute the point (1,0) into the circle equation:

\[ (1 + 2k)^2 + (0 - 3k)^2 = 10 \]

Expand each term:

\[ (1 + 2k)^2 = 1 + 4k + 4k^2 \] \[ (-3k)^2 = 9k^2 \]

Sum them:

\[ 1 + 4k + 4k^2 + 9k^2 = 10 \]

Combine like terms:

\[ 13k^2 + 4k + 1 = 10 \]

Move 10 to the left:

\[ 13k^2 + 4k - 9 = 0 \]

Solve the quadratic:

\[ k = \frac{-4 \pm \sqrt{4^2 - 4(13)(-9)}}{2(13)} \] \[ k = \frac{-4 \pm \sqrt{16 + 468}}{26} \] \[ k = \frac{-4 \pm \sqrt{484}}{26} \] \[ k = \frac{-4 \pm 22}{26} \]

Thus, the two possible values are:

\[ k = \frac{18}{26} = \frac{9}{13}, \qquad k = \frac{-26}{26} = -1 \]

Final Answer:

\[ \boxed{k = -1 \quad \text{or} \quad k = \frac{9}{13}} \]

Finding the Value of k for a Circle Passing Through a Given Point – Diagram

Answer

Farthest Distance from a Point to a Circle – Diagram

Step 1: Rewrite the circle in standard form

$$x^2 + y^2 - 16y = 0$$ Complete the square: $$x^2 + (y^2 - 16y + 64) = 64$$ $$(y - 8)^2 + x^2 = 8^2$$ The center is $(0,8)$ and the radius is $$r = 8.$$

Step 2: Find the distance from the point to the center

$$d_1 = \sqrt{(8 - 0)^2 + (10 - 8)^2}$$ $$d_1 = \sqrt{64 + 4} = \sqrt{68} = 8.25$$

Step 3: Farthest distance = center-to-point distance + radius

$$d_2 = d_1 + r$$ $$d_2 = 8.25 + 8 = 16.25$$

Final Answer:

$$\boxed{16.25}$$

Answer

Let the third vertex be $(x, y)$. Using the area formula with $(2,4)$ and $(-2,3)$:
$\text{Area} = \frac{1}{2}\left|2(3 - y) + (-2)(y - 4) + x(4 - 3)\right| = 2$
$\frac{1}{2}|x - 4y + 14| = 2 \Rightarrow |x - 4y + 14| = 4$
Taking the branch matching the choices:
$x - 4y + 14 = 4$
$\boxed{x - 4y + 10 = 0}$

Answer

Divide through by 2:
$x^2 + y^2 - \frac{3}{2}x + \frac{5}{2}y + 1 = 0$
Center $\left(\frac{3}{4}, -\frac{5}{4}\right)$; radius:
$r = \sqrt{\left(\tfrac{3}{4}\right)^2 + \left(\tfrac{5}{4}\right)^2 - 1} = \sqrt{\tfrac{9}{8}} = 1.0607$
Circumference:
$C = 2\pi r = 2\pi(1.0607)$
$\boxed{C = 6.67}$

Answer

The standard form $(x - h)^2 + (y - k)^2 = r^2$ gives center $(h, k)$ and radius $r$.
From $(x + 3)^2 + y^2 = 13$:
Center $= (-3, 0)$, $r = \sqrt{13} = 3.606$
$\boxed{\text{Center }(-3, 0),\ r = 3.606}$

Answer

Radius squared = distance from center $(-3, 4)$ to $(3, 6)$:
$r^2 = (3 + 3)^2 + (6 - 4)^2 = 36 + 4 = 40$
Equation: $(x + 3)^2 + (y - 4)^2 = 40$
Expanding:
$x^2 + 6x + 9 + y^2 - 8y + 16 = 40$
$\boxed{x^2 + y^2 + 6x - 8y - 15 = 0}$

Answer

For $x^2 + y^2 + Dx + Ey + F = 0$, the center is $\left(-\frac{D}{2}, -\frac{E}{2}\right)$.
Here $D = 14$ and $E = -6$:
$\left(-\frac{14}{2}, -\frac{-6}{2}\right) = (-7, 3)$
$\boxed{(-7,\ 3)}$

Answer

The midpoint of a chord and the center define the perpendicular distance from the center to the chord.
$d = \sqrt{\left(-\tfrac{11}{2} + 3\right)^2 + \left(-\tfrac{13}{2} + 4\right)^2} = \sqrt{(-2.5)^2 + (-2.5)^2} = \sqrt{12.5}$
Half-chord: $\sqrt{r^2 - d^2} = \sqrt{25 - 12.5} = \sqrt{12.5}$
Chord length:
$L = 2\sqrt{12.5}$
$\boxed{L = 7.1}$

Answer

Circle $x^2 + y^2 + 6x - 16y + 24 = 0$ has center $(-3, 8)$ and radius:
$r = \sqrt{3^2 + 8^2 - 24} = \sqrt{49} = 7$
Distance from $(12, 2)$ to the center:
$d = \sqrt{(12+3)^2 + (2-8)^2} = \sqrt{261} = 16.155$
Farthest distance to the circle:
$d + r = 16.155 + 7$
$\boxed{23.155}$

Answer

The radical axis is found by subtracting the circle equations (equal $x^2, y^2$ coefficients):
$(x^2 + y^2 - 3) - (x^2 + y^2 - 6x + 6y + 11) = 0$
$6x - 6y - 14 = 0$
$\boxed{3x - 3y - 7 = 0}$

Answer

First normalize the second circle (divide by 3):
$x^2 + y^2 - x + 2y - \frac{20}{3} = 0$
Subtract from the first circle:
$(x^2 + y^2 - 6x + 4y - 10) - (x^2 + y^2 - x + 2y - \tfrac{20}{3}) = 0$
$-5x + 2y - \frac{10}{3} = 0$
Multiply by $-3$:
$\boxed{15x - 6y + 10 = 0}$

Answer

Part 1.

For $x^2 + y^2 + 4x - 6y - 12 = 0$, the center is $\left(-\frac{D}{2}, -\frac{E}{2}\right) = \left(-\frac{4}{2}, -\frac{-6}{2}\right)$:
$\boxed{(-2,\ 3)}$

Part 2.

Radius: $r = \sqrt{2^2 + 3^2 + 12} = \sqrt{25} = 5$
Area:
$A = \pi r^2 = \pi (5)^2 = 25\pi$
$\boxed{A = 78.54}$

Part 3.

The tangent is perpendicular to the radius at the point of tangency.
Slope of the radius from center $(-2, 3)$ to $(2, 6)$:
$m_r = \frac{6 - 3}{2 - (-2)} = \frac{3}{4}$
Tangent slope (negative reciprocal):
$\boxed{-\frac{4}{3}}$

Answer

Part 1.

The condition is $\text{dist to }(-5,-6) = 2\,(\text{dist to }(2,-3))$:
$(x+5)^2 + (y+6)^2 = 4\left[(x-2)^2 + (y+3)^2\right]$
$x^2 + y^2 + 10x + 12y + 61 = 4x^2 + 4y^2 - 16x + 24y + 52$
Collecting terms:
$\boxed{3x^2 + 3y^2 - 26x + 12y - 9 = 0}$

Part 2.

Divide the locus by 3:
$x^2 + y^2 - \frac{26}{3}x + 4y - 3 = 0$
Center $\left(\frac{26}{6}, -2\right)$:
$\boxed{(4.33,\ -2)}$

Part 3.

From $x^2 + y^2 - \frac{26}{3}x + 4y - 3 = 0$, center $\left(\frac{13}{3}, -2\right)$, radius:
$r = \sqrt{\left(\tfrac{13}{3}\right)^2 + 2^2 + 3} = \sqrt{25.78} = 5.08$
Area:
$A = \pi r^2 = \pi(25.78)$
$\boxed{A = 80.98}$

Answer

The line through the intersection points is the radical axis, found by subtracting the circle equations:
$(x^2 + y^2 - 4x - 20y + 68) - (x^2 + y^2 - 20x - 8y + 52) = 0$
$16x - 12y + 16 = 0$
Divide by 4:
$\boxed{4x - 3y + 4 = 0}$

Answer

Part 1.

Rewrite: $x^2 + y^2 - 6x + 12y + 9 = 9 \Rightarrow x^2 + y^2 - 6x + 12y = 0$.
Complete the square: $(x-3)^2 + (y+6)^2 = 45$
Radius:
$r = \sqrt{45}$
$\boxed{r = 6.71}$

Part 2.

From $(x-3)^2 + (y+6)^2 = 45$, the center is:
$\boxed{(3,\ -6)}$

Answer

Replacing $x$ with $-x$ leaves $y = (-x)^2 + 1 = x^2 + 1$ unchanged, so the curve is symmetric about the y-axis. (Replacing $y$ with $-y$ does not give the same equation, so it is not symmetric about the x-axis or origin.)
$\boxed{\text{y-axis}}$

Answer

Let a family circle have center $(x, y)$ and radius $r$. Tangent to the x-axis means $r = y$. Tangent (externally) to the given circle (center $(4,8)$, radius 5) means the center distance equals $5 + r$:
$\sqrt{(x-4)^2 + (y-8)^2} = 5 + y$
Squaring:
$(x-4)^2 + (y-8)^2 = (5 + y)^2$
$x^2 - 8x + 16 + y^2 - 16y + 64 = 25 + 10y + y^2$
$\boxed{x^2 - 8x - 26y + 55 = 0}$

Answer

Part 1.

Rewrite as $x^2 + y^2 + 2x - 4y - 4 = 0$. The center is $\left(-\frac{D}{2}, -\frac{E}{2}\right) = \left(-\frac{2}{2}, -\frac{-4}{2}\right)$:
$\boxed{(-1,\ 2)}$

Part 2.

A diameter passes through the center $(-1, 2)$. Parallel to $3x + 5y = 4$ means the form $3x + 5y = C$:
$3(-1) + 5(2) = -3 + 10 = 7$
$\boxed{3x + 5y - 7 = 0}$

Part 3.

Let a family circle have center $(x, y)$ and radius $r$. Tangent to the Y-axis means $r = x$. Tangent (internally) to the given circle (center $(-1,2)$, radius 3) means the center distance equals $3 - r$:
$\sqrt{(x+1)^2 + (y-2)^2} = 3 - x$
Squaring:
$(x+1)^2 + (y-2)^2 = (3 - x)^2$
$2x + 1 + (y-2)^2 = 9 - 6x$
$\boxed{y^2 + 8x - 4y - 4 = 0}$

Answer

Slope of the tangent at $(3, 4)$ on $x^2 + y^2 = 25$: differentiating, $y' = -\frac{x}{y} = -\frac{3}{4}$.
The subtangent length is $\left|\frac{y_0}{y'}\right|$:
$\left|\frac{4}{-3/4}\right| = \frac{16}{3}$
$\boxed{\frac{16}{3}}$

Answer

The distance from an external point to the point of tangency equals the tangent length, $\sqrt{\text{power of the point}}$.
Substitute $A(12, 2)$ into the circle expression:
$12^2 + 2^2 + 6(12) - 16(2) + 24 = 144 + 4 + 72 - 32 + 24 = 212$
$L = \sqrt{212}$
$\boxed{L = 14.56}$

Answer

Rewrite in standard conic form $r = \frac{ed}{1 + e\cos\theta}$:
$r = \frac{3}{3 + 2\cos\theta} = \frac{1}{1 + \frac{2}{3}\cos\theta}$
The eccentricity is $e = \frac{2}{3} < 1$, which defines an ellipse.
$\boxed{\text{an ellipse}}$

Answer

Part 1.

Area $= \frac{1}{2}|\vec{AB} \times \vec{AC}|$, with $\vec{AB} = (2, -1, -4)$ and $\vec{AC} = (3, 2, 1)$:
$\vec{AB} \times \vec{AC} = (7, -14, 7)$
$|\vec{AB} \times \vec{AC}| = \sqrt{49 + 196 + 49} = \sqrt{294} = 17.15$
$A = \frac{1}{2}(17.15)$
$\boxed{A = 8.57}$

Part 2.

The distance from $C$ to line $AB$ follows from the triangle area: $A = \frac{1}{2}(AB)(d)$.
$AB = \sqrt{2^2 + 1^2 + 4^2} = \sqrt{21} = 4.583$
$d = \frac{2A}{AB} = \frac{2(8.57)}{4.583}$
$\boxed{d = 3.74}$

Answer

The normal is the cross product of two vectors in the plane.
$\vec{ab} = (3, 2, -5)$, $\vec{ac} = (1, 1, -1)$
$\vec{ab} \times \vec{ac} = \big(2(-1)-(-5)(1),\ -(3(-1)-(-5)(1)),\ 3(1)-2(1)\big)$
$= (3, -2, 1)$
$\boxed{3i - 2j + k}$

Answer

Part 1.

Let the center be $(h, 0)$ on the x-axis, equidistant from both points:
$(h+1)^2 + 1^2 = (h-1)^2 + 3^2$
$2h + 2 = -2h + 10 \Rightarrow 4h = 8$
$\boxed{(2,\ 0)}$

Part 2.

Radius squared from center $(2,0)$ to $(-1,1)$:
$r^2 = (2+1)^2 + (0-1)^2 = 10$
$(x-2)^2 + y^2 = 10$
Expand:
$\boxed{x^2 + y^2 - 4x - 6 = 0}$

Part 3.

Translate to the new origin $(-5, 7)$ via $x = x' - 5$, $y = y' + 7$ in $x^2 + y^2 - 4x - 6 = 0$:
$(x'-5)^2 + (y'+7)^2 - 4(x'-5) - 6 = 0$
$x'^2 - 10x' + 25 + y'^2 + 14y' + 49 - 4x' + 20 - 6 = 0$
$\boxed{x'^2 + y'^2 - 14x' + 14y' + 88 = 0}$

Answer

For $y = a\sin(bx + c) + d$: amplitude $= |a|$ and period $= \frac{2\pi}{|b|}$.
Increasing $a$ increases the amplitude. The period is inversely related to $b$, so increasing $b$ would shorten the period — $b$ must not be increased. Thus the only quantity to increase is $a$.
$\boxed{a \text{ only}}$

Answer

Part 1.

For $f(t) = A\sin(Bt)$, the amplitude is $|A|$.
Here $A = 3$:
$\boxed{3}$

Part 2.

Period $= \frac{2\pi}{B}$ with $B = 2$:
$\frac{2\pi}{2}$
$\boxed{\pi}$

Part 3.

Frequency is the reciprocal of the period:
$f = \frac{1}{T} = \frac{1}{\pi}$
$\boxed{\frac{1}{\pi}}$

Answer

Part 1.

For $y = A\sin(Bt)$ the period is $T = \dfrac{2\pi}{B}$. Here $B = \tfrac{3}{2}$:
$$T = \frac{2\pi}{3/2} = \boxed{\frac{4\pi}{3}}$$

Part 2.

Frequency is the reciprocal of the period:
$$f = \frac{1}{T} = \frac{1}{4\pi/3} = \boxed{\frac{3}{4\pi}}$$

Part 3.

The amplitude is the coefficient $A$ multiplying the sine:
$$A = \boxed{\frac{\pi}{2}}$$

Answer

Check the y-intercept by setting $x = 0$:
$y = \frac{0 + 1}{0 - 1} = -1$
So the curve does have a y-intercept at $(0, -1)$, making the statement "it has no y-intercept" false. (The others are true: no real x-intercept since $x^2+1 \ne 0$; vertical asymptotes at $x = \pm 1$; even function, so symmetric about the y-axis.)
$\boxed{\text{it has no y-intercept}}$

Answer

Using $x^2 + y^2 = r^2$ and $y = r\sin\theta$:
$r^2 = 4r\sin\theta$
$\boxed{r = 4\sin\theta}$

Question 2

Problem: Equation of a Circle with Given Center and Tangent to a Given Line

Find the equation of the circle with center at (-3,8) and tangent to the line x-y+5=0.

Question 3

Problem: Length of the Tangent to a Circle

Determine the length of the tangent to the circle x²+y²-4x-5=0 from (8,-2).

Question 4

Locus of a Moving Point that Forms a Triangle of Specific Area

Find the locus of a moving point which forms a triangle of area 21 square units with the point (2, -7) and (-4, 3).
Ans. 5x+3y+32=0 or 5x+3y-10=0

Question 5

Equation of the Curve Connecting Centers of Circles Tangent to the Fixed Circle and the x-axis

There is a fixed circle having a radius of 6 with center at (10, 12). Find the equation of the curve connecting the centers of all circles that are tangent to the fixed circle and the x axis.

Question 6

Common Tangent of Two Circles (CE Board May 2015)

Given the equations of two circles:

C₁: x² + y² + 2x + 4y − 3 = 0
C₂: x² + y² − 8x − 6y + 7 = 0

Determine the equation of the common tangent at their point of contact.

Question 7

Finding the y-coordinate of the Point of Tangency

A circle has its center at (2, 3). If the circle is tangent to the y-axis and a tangent line has its point of tangency at P(3, y), determine the value of y.

Question 8

Finding the Value of k for a Circle Passing Through a Given Point

For what value of k does the circle $ (x + 2k)^2 + (y - 3k)^2 = 10 $ pass through the point (1,0)?

Question 9

Farthest Distance from a Point to a Circle

Determine the farthest distance from the point $(8,10)$ to the circle $x^2 + y^2 = 16y.$

Question 10

Question Bank: t641

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

Two vertices of a triangle are at (2, 4) and (-2, 3) and the area is 2 square units. The locus of the third vertex is:

x + 4y - 10 = 0
x - 4y + 10 = 0
x - 2y - 5 = 0
x + 2y - 5 = 0

Question 11

Question Bank: t646

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

What is the circumference of the circle 2x^2 + 2y^2 - 3x + 5y + 2 = 0.

7.21
3.58
5.24
6.67

Question 12

Question Bank: t647

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

Given the circle: (x + 3)^2 + y^2 = 13. Find the center and radius.

Center (3, 0), radius = 3.742
Center (-3, 0), radius = 3.742
Center (3, 0), radius = 3.606
Center (- 3, 0), radius = 3.606

Question 13

Question Bank: t648

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

What is the equation of a circle with center at (-3, 4) and passing through (3, 6).

x^2 + y^2 + 6x - 8y - 15 = 0
x^2 + y^2 + 8x - 6y - 15 = 0
x^2 + y^2 - 6x + 8y - 19 = 0
x^2 + y^2 - 6x - 8y - 19 = 0

Question 14

Question Bank: t649

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

Where is the center of the circle x^2 + y^2 + 14x - 6y + 22 = 0?

(3, -7)
(-3, 7)
(-7, 3)
(7, -3) "x-intercept," but the problem's solution actually calculates the y-intercept, which is -5/2. The correct option A matches this value.) set of questions (31–45)!

Question 15

Question Bank: t650

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

The radius of a circle is 5 and its center is at (-3, -4). Find the length of chord that is bisected at (-11/2, -13/2).

7.1
5.2
8.3
6.4

Question 16

Question Bank: t651

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

Find the farthest distance from the point (12, 2) to the circle x^2 + y^2 + 6x - 16y + 24 = 0.

16.155
18.155
23.155
9.155

Question 17

Question Bank: t652

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

What is the equation of the radical axis of the circles x^2 + y^2 = 3 and x^2 + y^2 - 6x + 6y + 11 = 0?

3x + 2y - 2 = 0
3x - 3y - 7 = 0
3x - 3y - 2 = 0
3x + 3y + 2 = 0

Question 18

Question Bank: t653

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

Find the equation of the radical axis of the circles x^2 + y^2 - 6x + 4y - 10 = 0 and 3x^2 + 3y^2 - 3x + 6y - 20 = 0.

6x + 15y + 10 = 0
6x - 15y + 10 = 0
15x - 6y + 10 = 0
15x + 6y + 10 = 0

Question 19

Question Bank: t654

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

Given the circle x^2 + y^2 + 4x - 6y - 12 = 0.

What is the location of the center of the circle?

(-2, 3)
(2, -3)
(2, 3)
(-2, - 3)

Find the area of the circle.

157.08
78.54
83.87
132.98

Find the slope of the line tangent to the circle at (2, 6).

-3/4
3/4
-4/3
4/3

Question 20

Question Bank: t657

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

A point P(x, y) moves such that it is always twice as far from (-5, -6) as it is from (2, -3).

What is the equation of the locus of P.

3x^2 + 3y^2 - 26x + 12y - 9 = 0
3x^2 + 3y^2 + 26x - 12y - 9 = 0
4x^2 + 3y^2 + 26x - 12y + 9 = 0
4x^2 - 3y^2 + 26x - 12y - 9 = 0

Where is the center of the locus of P?

(4.33, -2)
(5.43, -2)
(3.23, 2)
(3.67, 2)

What is the area bounded by the locus of the point?

78.32
80.98
87.65
102.34

Question 21

Question Bank: t660

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

Find the equation of the line joining the points of intersection of the circles x^2 + y^2 - 4x - 20y + 68 = 0 and x^2 + y^2 - 20x - 8y + 52 = 0.

4x - 3y + 4 = 0
4x + 3y + 4 = 0
4x + 3y - 4 = 0
4x - 3y - 4 = 0

Question 22

Question Bank: t661

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

Given the circle x^2 + y^2 - 6x + 12y + 9 = 9.

What is the radius of the circle?

7.21
6.71
6.28
8.63

Where is the center of the circle?

(3, - 6)
(-3, -6)
(-3, 6)
(3, 6)

Find the shortest distance from the line y = 2x + 10 to the center of the circle.

6.87
10.24
8.49
9.84

Question 23

Question Bank: t667

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

The curve $y = x^2 + 1$ is symmetric with respect to:

the origin
x & y - axes
x - axis
y - axis

Question 24

Question Bank: t676

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

A circle with radius of 5 has its center at (4, 8). Find the equation of the centers of family of circles tangent to the given circle and the X- axis.

$y^2 - 4x - 26y + 55 = 0$
$x^2 - 8x - 26y + 55 = 0$
$x^2 - 9x - 24y + 55 = 0$
$x^2 + 8x - 20y + 64 = 0$

Question 25

Question Bank: t677

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

A circle is defined by the equation $x^2 + y^2 + 2x - 4y = 4$.

The center of the circle is at:

(- 1, -2)
(1, 2)
(1, -2)
(- 1, 2)

Find the equation of the diameter of the circle that is parallel to the line $3x + 5y = 4$.

$3x + 5y - 6 = 0$
$3x + 5y - 8 = 0$
$3x + 5y - 7 = 0$
$3x + 5y - 3 = 0$

Find the equation of the centers of family of circles that is tangent to the given circle and the Y-axis.

$y^2 - 2x - 3y - 7 = 0$
$y^2 + 2x - 4y - 2 = 0$
$y^2 - 8x - 4y - 5 = 0$
$y^2 + 8x - 4y - 4 = 0$ set of questions (61–75)!

Question 26

Question Bank: t715

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

What is the equation of the tangent to the curve $y = 3x^3 - 15$ at $x = 2$?

$81x - y + 153 = 0$
$81x - y + 162 = 0$
$81x - y - 153 = 0$
$81x + 2y - 153 = 0$

Solution pending in psadquestions/t715.json.

Question 27

Question Bank: t716

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

A line is tangent to the circle $x^2 + y^2 = 25$ at (3, 4), find the length of the sub-tangent.

3
16/3
3/16
4

Question 28

Question Bank: t717

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

A line tangent to the circle $x^2 + y^2 + 6x - 16y + 24 = 0$ passes through point A(12, 2). Find the distance from the point of tangency to point A.

14.56
13.21
15.58
15.11

Question 29

Question Bank: t721

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

The polar curve $r = 3 / (3 + 2\cos\theta)$ is:

a circle
an ellipse
a parabola
a hyperbola

Question 30

Question Bank: t734

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

The points A(1, 0, -1), B(3, -1, -5), and C(4, 2, 0) are vertices of a triangle.

What is the area of the triangle?

8.57
9.67
4.87
12.49

What is the distance from point C to the line through A and B?

1.23
2.89
3.74
4.32

Question 31

Question Bank: t739

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

Find the vector that is perpendicular to the plane passing through the points a(1, 2, 6), b(4, 4, 1), and c(2, 3, 5).

3i - 2j - k
2i + 3j + k
3i - 2j + k
2i + 3j - k

Question 32

Question Bank: t744

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

The graph of the function $f(x) = x^2$ is translated 3 units to the left and reflected over the x-axis. If the resulting function is represented by $g(x)$, what is the value of g(5)?

-64
-9
-49
-54

Solution pending in psadquestions/t744.json.

Question 33

Question Bank: t745

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

A circle passing through (-1, 1) and (1, 3) have its center on the x-axis.

Where is the center of the circle?

(3, 0)
(2, 0)
(-1, 0)
(-2, 0)

What is the equation of the circle?

$x^2 + y^2 - 4x - 10 = 0$
$x^2 + y^2 + 4x - 8 = 0$
$x^2 + y^2 - 4x - 6 = 0$
$x^2 + y^2 + 4x - 11 = 0$

What is the equation of the circle with respect to translated axes whose origin is at coordinates is (-5, 7)?

$x'^2 + y'^2 - 14x' + 14y' + 88 = 0$
$x'^2 + y'^2 - 14x' - 14y' - 88 = 0$
$x'^2 + y'^2 + 14x' + 14y' + 88 = 0$
$x'^2 + y'^2 + 14x' - 14y' + 88 = 0$

Question 34

Question Bank: t752

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

Given the sinusoidal curve y = a sin (bx + c) + d. Which of the following must be increased to increase the amplitude & period of the curve?

d only
b only
a and b
a only

Question 35

Question Bank: t753

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

Given the sinusoidal function f(t) = 3 sin 2t. Determine the following:

The amplitude of the curve.

2
3
1
undefined

The period of the curve.

π
2π
π/2
π/3

The frequency of the curve.

undefined
2/π
1/π
3

Question 36

Question Bank: t756

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

Given the curve y = (π/2) sin (3t/2).

What is the period of the curve?

3π/2
2π/3
4π
4π/3

What is the frequency of the curve?

1/4π
1/3π
3/4π
4/3π

What is the amplitude of the curve?

π/4
π/2
π/8
π/6

Question 37

Question Bank: t759

MSTE - Analytic Geometry / Analytic Geometry / Gemini mapped Chapter 4 to 6

Given the curve: y = (x^2 + 1) / (x^2 - 1). Which of the following is NOT true?

it has no y-intercept
it has no x-intercept
it has two vertical asymptotes
it is symmetrical to the y-axis

Question 38

Question Bank: w32

MSTE - Analytic Geometry / Polar Equation / MSTE May 2019

Find a polar equation that has the same graph as the circle $x^2 + y^2 = 4y$.

$r = 4\cos\theta$
$r = 4\sin\theta + \cos\theta$
$r = 4(\sin\theta + \cos\theta)$
$r = 4\sin\theta$

Circle

Problem: Equation of a Circle with Given Center and Tangent to a Given Line

Problem: Length of the Tangent to a Circle

Locus of a Moving Point that Forms a Triangle of Specific Area

Equation of the Curve Connecting Centers of Circles Tangent to the Fixed Circle and the x-axis

Common Tangent of Two Circles (CE Board May 2015)

Finding the y-coordinate of the Point of Tangency

Finding the Value of k for a Circle Passing Through a Given Point

Farthest Distance from a Point to a Circle

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