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Conic Sections

Conic Sections — Definitions, Locus, and General Equation

General Second-Degree Equation of a Conic

Any conic section can be represented by the general quadratic equation:

$$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$

The presence of the $Bxy$ term indicates a possible rotation of axes. Circles, ellipses, parabolas, and hyperbolas can all appear rotated when $B \neq 0$.

How to Distinguish Conics Using A, B, and C

Ignoring rotation (or after rotating axes so that $B = 0$), the nature of the conic is determined by the signs of $A$ and $C$:

With rotation included ($B \neq 0$), the classification is done using the discriminant:

$$ \Delta = B^2 - 4AC $$

Eccentricity of Conic Sections

The eccentricity of a conic section describes the shape and “flatness” of the curve. It is defined as:

$$ e = \frac{\text{distance to focus}}{\text{distance to directrix}} $$

This value fully determines the nature of the conic:

1. Eccentricity of an Ellipse

For the ellipse $$ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1, $$ the first eccentricity is:

$$ e = \frac{c}{a} \qquad \text{where } c^2 = a^2 - b^2 $$

The eccentricity describes flatness:

2. Second Eccentricity of an Ellipse

An alternate measure of shape is the second eccentricity:

$$ e' = \frac{c}{b} $$

Using $c^2 = a^2 - b^2$, we may also write:

$$ e' = \sqrt{\frac{a^2}{b^2} - 1} $$

3. Distance From Center to the Directrix of an Ellipse

The directrices of the ellipse lie at the distance:

$$ d = \frac{a^2}{c} = \frac{a}{e} $$

measured horizontally from the center.

Derivation

Start with the eccentricity definition:

$$ e = \frac{c}{a}. $$

A point $P$ on the ellipse satisfies:

$$ \frac{\text{distance to focus}}{\text{distance to directrix}} = e. $$

Evaluating this at the vertex $(a,0)$ gives:

$$ \frac{a - c}{\frac{a^2}{c} - a} = \frac{c}{a}, $$

which simplifies to confirm the directrix location:

$$ x = \frac{a^2}{c}. $$

Thus:

$$ d = \frac{a^2}{c} = \frac{a}{e}. $$

Exam Generator Problems

Additional board-style practice items for this topic.

Question Bank: q407

MSTE - Analytic Geometry / Conic Sections / Engr. Janclyde Espinosa (Clidez)

The curve x2+y2-4x+2y-4=0 has its center at?

Answer:

  1. (2,-1)
  2. (2,1)
  3. (-2,-1)
  4. (-2,1)
Complete the square:
$x^2-4x+y^2+2y-4=0$
$(x-2)^2+(y+1)^2=9$
The center is:
$\boxed{(2,-1)}$

Question Bank: q408

MSTE - Analytic Geometry / Conic Sections / Engr. Janclyde Espinosa (Clidez)

What is the radius of the curve x2+y2-6x-8y-11=0?

Answer:

  1. 6
  2. 11
  3. 1
  4. 3
Complete the square:
$x^2-6x+y^2-8y-11=0$
$(x-3)^2+(y-4)^2=36$
$\boxed{r=6}$

Question Bank: q414

MSTE - Analytic Geometry / Conic Sections / Engr. Janclyde Espinosa (Clidez)

A circle passes through the point (5,7) and has its center at (2,3). Find its equation.

Answer:

  1. (x-2)2+(y-3)2=25
  2. (x+2)2+(y-3)2=25
  3. (x-2)2+(y+3)2=25
  4. (x+2)2+(y+3)2=25
The radius is the distance from center $(2,3)$ to point $(5,7)$:
$r=\sqrt{(5-2)^2+(7-3)^2}=5$
Circle equation:
$\boxed{(x-2)^2+(y-3)^2=25}$

Question Bank: q415

MSTE - Analytic Geometry / Conic Sections / Engr. Janclyde Espinosa (Clidez)

How far is the center of the circle x2+y2-6x-8y-11=0 from the y-axis?

Answer:

  1. 3
  2. 4
  3. 6
  4. 2
Complete the square:
$x^2-6x+y^2-8y-11=0$
$(x-3)^2+(y-4)^2=36$
The center is $(3,4)$, so its distance from the y-axis is $|3|$.
$\boxed{3}$

Question Bank: q416

MSTE - Analytic Geometry / Conic Sections / Engr. Janclyde Espinosa (Clidez)

Find the vertex of the parabola whose equation is y2-2x-4y+2=0.

Answer:

  1. (-1,2)
  2. (1,-2)
  3. (2,-1)
  4. (2,1)
Complete the square in $y$:
$y^2-4y-2x+2=0$
$(y-2)^2=2x+2=2(x+1)$
The vertex is:
$\boxed{(-1,2)}$

Question Bank: q417

MSTE - Analytic Geometry / Conic Sections / Engr. Janclyde Espinosa (Clidez)

Determine the equation of the directrix of the parabola: y2=16x

Answer:

  1. x=-4
  2. x=4
  3. y=-4
  4. y=4
For $y^2=4px$, compare with $y^2=16x$:
$4p=16$, so $p=4$. The directrix is:
$\boxed{x=-4}$

Question Bank: q418

MSTE - Analytic Geometry / Conic Sections / Engr. Janclyde Espinosa (Clidez)

A parabola has an equation of x2-4y-2x+8=0. Find the length of the latus rectum.

Answer:

  1. 4
  2. 1
  3. 16
  4. 8
Complete the square:
$x^2-2x-4y+8=0$
$(x-1)^2=4y-7=4(y-7/4)$
Thus $4p=4$, so $p=1$. The latus rectum length is $4p$.
$\boxed{4}$

Question Bank: q482

MSTE - Analytic Geometry / Conic Sections / Engr. Janclyde Espinosa (Clidez)

Given the curve y2+8x+4y-28=0, determine the length of the latus rectum.

Answer:

  1. 8
  2. 6
  3. 4
  4. 2
Complete the square:
$y^2+4y+8x-28=0$
$(y+2)^2=-8(x-4)$
Compare with $(y-k)^2=4p(x-h)$, so $4p=-8$. The latus rectum length is $|4p|$.
$\boxed{8}$

Question Bank: q483

MSTE - Analytic Geometry / Conic Sections / Engr. Janclyde Espinosa (Clidez)

It is the locus of all points equidistant from a fixed point and a fixed line.

Answer:

  1. Parabola
  2. Ellipse
  3. Circle
  4. Hyperbola
The locus of points equidistant from a fixed point called the focus and a fixed line called the directrix is a parabola.
$\boxed{\text{Parabola}}$

Question Bank: q484

MSTE - Analytic Geometry / Conic Sections / Engr. Janclyde Espinosa (Clidez)

It is a conic section with an eccentricity greater than 1.

Answer:

  1. Hyperbola
  2. Ellipse
  3. Circle
  4. Parabola
Conic eccentricity rules: ellipse has $e<1$, parabola has $e=1$, and hyperbola has $e>1$.
$\boxed{\text{Hyperbola}}$

Question Bank: q487

MSTE - Analytic Geometry / Conic Sections / Engr. Janclyde Espinosa (Clidez)

Determine the area enclosed by the curve x2-10x+4y+y2=196

Answer:

  1. 225π
  2. 144π
  3. 15π
  4. 12π
Complete the square:
$x^2-10x+y^2+4y=196$
$(x-5)^2+(y+2)^2=225$
The radius is 15, so area is:
$A=\pi(15)^2$
$\boxed{225\pi}$

Question Bank: q488

MSTE - Analytic Geometry / Conic Sections / Engr. Janclyde Espinosa (Clidez)

Determine the distance between the foci of the following ellipse:

q488

Answer:

  1. 4
  2. 2
  3. 3
  4. 8

Solution pending in psadquestions/q488.json.

Question Bank: q489

MSTE - Analytic Geometry / Conic Sections / Engr. Janclyde Espinosa (Clidez)

The following hyperbola has an asymptote of:

q489

Answer:

  1. y+2=3/2(x-2)
  2. y+2=-3/4(x-2)
  3. y+3=-3/4(x-2)
  4. y+2=2/3(x-2)

Solution pending in psadquestions/q489.json.

Question Bank: q490

MSTE - Analytic Geometry / Conic Sections / Engr. Janclyde Espinosa (Clidez)

A construction worker built a parabolic hut which is 16.1 m wide at the base and 12.4 m high at the center. How high above the base should a ceiling 12.2 m wide be constructed?

Answer:

  1. 5.28
  2. 7.12
  3. 6.86
  4. 4.32
Model the hut as a downward parabola with vertex height 12.4 m and half-width 8.05 m at the base:
$y=12.4\left(1-\frac{x^2}{8.05^2}\right)$
A 12.2-m ceiling has half-width $x=6.1$ m:
$y=12.4\left(1-\frac{6.1^2}{8.05^2}\right)$
$\boxed{5.28\text{ m}}$

Question Bank: t684

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

What conic section is represented by $8x^2 - 4xy + 5x = 10$?

  1. a parabola
  2. a hyperbola
  3. an ellipse
  4. a circle

Solution pending in psadquestions/t684.json.

Question Bank: t687

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

The eccentricity of an ellipse is 0.6 and the distance between its foci is 6 units. Find the distance between its directrices.

  1. 8.33
  2. 15.67
  3. 16.67
  4. 21.33
Distance between foci $2c = 6 \Rightarrow c = 3$. Eccentricity $e = \frac{c}{a} = 0.6 \Rightarrow a = \frac{3}{0.6} = 5$.
Directrices are at $x = \pm\frac{a}{e}$, so their separation is:
$\frac{2a}{e} = \frac{2(5)}{0.6}$
$\boxed{16.67}$

Question Bank: t694

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

Find the distance from the end of the major axis to the end of the minor axis of the ellipse $9x^2 + 25y^2 - 225 = 0$.

  1. 5.83
  2. 4.27
  3. 6.32
  4. 2.00 numbers the sub-questions as 2, 3, and 3 instead of 1, 2, and 3. This has been transcribed exactly as it appears in the source material.)
Rewrite: $\frac{x^2}{25} + \frac{y^2}{9} = 1$, so $a = 5$ (major) and $b = 3$ (minor).
End of major axis $(5, 0)$, end of minor axis $(0, 3)$. Distance:
$d = \sqrt{5^2 + 3^2} = \sqrt{34}$
$\boxed{d = 5.83}$

Question Bank: t702

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

What conic section is represented by 2x^2 - 4xy + 5x = 10?

  1. a parabola
  2. a hyperbola
  3. an ellipse
  4. a circle
Identify the discriminant $B^2 - 4AC$ with $A = 2$, $B = -4$, $C = 0$:
$B^2 - 4AC = (-4)^2 - 4(2)(0) = 16 > 0$
A positive discriminant means the conic is a hyperbola.
$\boxed{\text{a hyperbola}}$

Question Bank: t740

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

A sphere of radius 15 units have its center at the origin. Given three points P1(6, 8, 9), P2(5, 8, 10), and P3(4, 3, 12). Which of the following is/are inside the sphere.

  1. P2 and P3 only
  2. P1 only
  3. P1, P2, and P3
  4. P1 and P2 only
A point is inside the sphere if $x^2 + y^2 + z^2 < R^2 = 225$.
$P_1: 36 + 64 + 81 = 181$
$P_2: 25 + 64 + 100 = 189$
$P_3: 16 + 9 + 144 = 169$
All are less than 225, so all three lie inside.
$\boxed{P_1, P_2, \text{ and } P_3}$

Question Bank: t741

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

A sphere of radius 6 units have its center at the origin. Which of the following points is outside the sphere?

  1. (3, 1, 4)
  2. (2, 5, 1)
  3. (3, 4, 3)
  4. (2, 4, 5)
A point is outside the sphere if $x^2 + y^2 + z^2 > R^2 = 36$.
$(3,1,4): 9+1+16 = 26$
$(2,5,1): 4+25+1 = 30$
$(3,4,3): 9+16+9 = 34$
$(2,4,5): 4+16+25 = 45 > 36$
$\boxed{(2,\ 4,\ 5)}$

Question Bank: t2102

MSTE - Analytic Geometry / Conic Sections / Besavilla CE Pre-Board Math & Surveying

Find the equation of the circle whose center is on the x-axis and which passes through the points (1, 3) and (4, 6).

  1. $x^2 + y^2 + 14x + 3 = 0$
  2. $x^2 + y^2 - 12x + 4 = 0$
  3. $x^2 + y^2 - 14x + 5 = 0$
  4. $x^2 + y^2 + 14x + 4 = 0$
  5. $x^2 + y^2 - 14x + 4 = 0$
Let the center be $(h,0)$ since it lies on the x-axis. Equal distances to $(1,3)$ and $(4,6)$ give
$(1-h)^2+3^2=(4-h)^2+6^2$
$h=7$
Radius squared:
$R^2=(1-7)^2+3^2=45$
Circle equation:
$(x-7)^2+y^2=45$
$\boxed{x^2+y^2-14x+4=0}$

Question Bank: t2104

MSTE - Analytic Geometry / Conic Sections / Besavilla CE Pre-Board Math & Surveying

Find the equations of the tangents to the circle $x^2 + y^2 = 5$ which make an angle of 45° with the x-axis.

  1. $x + y = \pm\sqrt{10}$
  2. $x - y = \pm\sqrt{10}$
  3. $x - y = \frac{\sqrt{10}}{2}$
  4. $x^2 - y^2 = \pm\sqrt{10}$
  5. $x + y = \pm 5$
A line making $45^\circ$ with the x-axis has slope 1, so write it as $y=x+b$, or $x-y+b=0$.
For tangency to $x^2+y^2=5$, the distance from the center $(0,0)$ to the line must equal the radius $\sqrt5$.
$\frac{|b|}{\sqrt{1^2+(-1)^2}}=\sqrt5$
$|b|=\sqrt{10}$
Thus $x-y=\pm\sqrt{10}$.
$\boxed{x-y=\pm\sqrt{10}}$

Question Bank: t2106

MSTE - Analytic Geometry / Conic Sections / Besavilla CE Pre-Board Math & Surveying

Find the equation of the hyperbola with center (1, 3), vertex (4, 3) and end of conjugate axis (1, 1).

  1. $4x^2 - 3y^2 - 4x + 50y - 113 = 0$
  2. $4x^2 - 9y^2 - 8x + 54y - 113 = 0$
  3. $4x^2 - 6y^2 - 6x + 64y - 113 = 0$
  4. $4x^2 - 9y^2 - 10x + 58y - 113 = 0$
  5. $4x^2 - 9y^2 - 9x + 56y - 113 = 0$
The center is $(1,3)$. Since the vertex is $(4,3)$, the transverse axis is horizontal and $a=3$. The end of the conjugate axis is $(1,1)$, so $b=2$.
Standard form:
$\frac{(x-1)^2}{9}-\frac{(y-3)^2}{4}=1$
Multiply by 36:
$4(x-1)^2-9(y-3)^2=36$
Expand and simplify:
$\boxed{4x^2-9y^2-8x+54y-113=0}$

Question Bank: t2107

MSTE - Analytic Geometry / Conic Sections / Besavilla CE Pre-Board Math & Surveying

A circular rotonda passes through the three points A(-4, 3), B(2, 1) and C(-2, -5). Determine the radius of this circular rotonda.

  1. 5.58
  2. 3.26
  3. 6.15
  4. 4.87
  5. 4.27
Use the circle form $x^2+y^2+Dx+Ey+F=0$ and substitute the three points.
For $A(-4,3)$: $25-4D+3E+F=0$
For $B(2,1)$: $5+2D+E+F=0$
For $C(-2,-5)$: $29-2D-5E+F=0$
Solving gives $D=\frac{42}{11}$ and $E=\frac{16}{11}$, so the center is $\left(-\frac{21}{11},-\frac{8}{11}\right)$.
Radius using point $B$:
$R=\sqrt{\left(2+\frac{21}{11}\right)^2+\left(1+\frac{8}{11}\right)^2}$
$\boxed{R\approx4.27}$

Question Bank: t2170

MSTE - Analytic Geometry / Conic Sections / Besavilla CE Pre-Board Math & Surveying

Determine the equation of the radical axis of the circles $x^2 + y^2 - 18x - 14y + 121 = 0$ and $x^2 + y^2 - 6x + 6y + 14 = 0$.

  1. $12x + 20y - 107 = 0$
  2. $12x + 20y + 107 = 0$
  3. $12x - 20y - 107 = 0$
  4. $12x - 20y + 107 = 0$
  5. $12x - 20y + 105 = 0$
For the radical axis, subtract the second circle equation from the first so the $x^2$ and $y^2$ terms cancel.
$(x^2+y^2-18x-14y+121)-(x^2+y^2-6x+6y+14)=0$
$-12x-20y+107=0$
Multiplying by $-1$ gives
$\boxed{12x+20y-107=0}$

Question Bank: w30

MSTE - Analytic Geometry / Conic Sections / MSTE May 2019

The equation of a circle is $x^2 + y^2 + 2ky = 0$. What is the value of $k$ when the length of the tangent from $(5, 4)$ to the circle is 1?

  1. -3
  2. -2
  3. -4
  4. -5
Complete the square: $x^2 + (y+k)^2 = k^2$, center $(0, -k)$, radius $r = k$.
Tangent-length relation: $d^2 = 1^2 + k^2$, where $d$ is the distance from the center to $(5,4)$.
$d^2 = (5-0)^2 + (4+k)^2 = 41 + 8k + k^2$
$1 + k^2 = 41 + 8k + k^2 \Rightarrow \boxed{k = -5}$

Question Bank: w31

MSTE - Analytic Geometry / Conic Sections / MSTE May 2019

A parabolic dish is in the form of a paraboloid 12 m in diameter and 2 m deep. Find the distance of the receiver from the vertex of the paraboloid.

  1. 2.6
  2. 4.5
  3. 6.25
  4. 4.0
With the vertex at the origin and $y^2 = 4ax$, the rim point is $(2, 6)$ (depth 2 m, radius 6 m):
$6^2 = 4a(2) \Rightarrow a = 4.5$
The receiver sits at the focus: $\boxed{a = 4.5\text{ m}}$
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