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Equation of the Diameter of a Conic

The diameter of a conic is the locus of the midpoints of a family of parallel chords drawn to the conic.

Equation of the Diameter of a Conic – Diagram

For any conic, parallel chords share a common direction. Let the slope of these parallel chords be $m$.

To find the equation of the diameter corresponding to slope $m$:

1. Differentiate the conic equation to obtain $y'$.
2. Substitute $y' = m$ into the derivative.
3. The resulting equation represents the locus of midpoints — the diameter.

Geometrically, each chord midpoint lies on a straight line (the diameter), and every chord in the family is parallel to the others.

Summary: The diameter associated with slope $m$ is determined by enforcing $$ y' = m $$ on the conic equation. This produces a linear locus describing all midpoint positions of the parallel chords.

Diameter of an Ellipse from Chords of Equal Slope

The chords of the ellipse $64x^2 + 25y^2 = 1600$ having equal slopes of $1/5$ are bisected by its diameter. Determine the equation of this diameter.

Diameter of an Ellipse from Chords of Equal Slope – Diagram

Differentiation:

$$ 64x^2 + 25y^2 = 1600 $$ Differentiate: $$ 128x + 50y\,y' = 0 $$

Substitute the slope of the parallel chords: $y' = \frac{1}{5}$

$$ 128x + 50y\left(\frac{1}{5}\right) = 0 $$ Simplify: $$ 128x + 10y = 0 $$ Final equation of the diameter: $$ \boxed{64x + 5y = 0} $$

Diameter of a Parabola from Chords Parallel to a Given Line

A parabola has equation $y^2 = 8x$. Find the equation of the diameter that bisects all chords parallel to the line $x - y = 4$.

Diameter of a Parabola from Chords Parallel to a Given Line – Diagram

Chords parallel to the line $x - y = 4$ have slope $$ y' = 1 $$

Differentiate the parabola:

$$ y^2 = 8x $$ $$ 2y\,y' = 8 $$

Substitute the slope of the parallel chords:

$$ 2y(1) = 8 $$ Thus, $$ \boxed{y = 4} $$

This horizontal line is the diameter that bisects all chords parallel to $x - y = 4$.

Problem: Diameter Through Midpoints of Parallel Chords

For the circle x2 + y2 = 25, find the diameter that bisects chords whose slope is 2.

The radius to the midpoint of a chord is perpendicular to the chord.

$$m_d=-\frac{1}{2}$$

Answer: The required diameter is $y=-\dfrac{x}{2}$.

Problem: Conjugate Diameter of an Ellipse

For the ellipse x2/36 + y2/16 = 1, find the diameter conjugate to chords with slope 1.

For $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, the conjugate diameter slope is $m_d=-\dfrac{b^2}{a^2m}$.

$$m_d=-\frac{16}{36(1)}=-\frac{4}{9}$$

Answer: The conjugate diameter is $y=-\dfrac{4}{9}x$.

Problem: Parabola Diameter for Parallel Chords

For y2 = 12x, find the diameter that bisects chords with slope 3.

For $y^2=4ax$, the midpoint locus of parallel chords with slope $m$ is $y=\dfrac{2a}{m}$.

$$4a=12 \Rightarrow a=3,\qquad y=\frac{2(3)}{3}=2$$

Answer: The diameter is the horizontal line $y=2$.

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Exam Generator Problems

Additional board-style practice items for this topic.

Question Bank: q702

MSTE - Analytic Geometry / Equation of the Diameter / Engr. Janclyde Espinosa (Clidez)

A parabola has the equation y2 = 8x Find the equation of the diameter of the parabola which bisects chords parallel to the line x-y=4.

  1. y = 4
  2. y = 2
  3. y = 3
  4. y = 1
For $y^2=4ax$, $4a=8$ so $a=2$. The diameter bisecting chords of slope $m$ is $y=2am$. Chords parallel to $x-y=4$ have slope $m=1$. Therefore:
$y=2(2)(1)$
$\boxed{y=4}$

Question Bank: t691

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

Given the ellipse $4x^2 + 9y^2 = 144$.

What is the equation of the ellipse with respect to the pole whose coordinate is (-6, 4)?

  1. $4x^2 + 9y^2 + 32x + 72y + 144 = 0$
  2. $4x^2 + 9y^2 + 32x - 64y + 144 = 0$
  3. $4x^2 + 9y^2 - 48x + 72y + 144 = 0$
  4. $4x^2 + 9y^2 - 48x - 56y - 144 = 0$

What is the equation of the diameter bisecting the chords having equal slope of -1/3?

  1. $3x - 4y = 0$
  2. $4x - 3y = 0$
  3. $3x + 4y = 0$
  4. $4x + 3y = 0$

What is the equation of the diameter bisecting the chords having equal slope of 1/2?

  1. $8x + 9y = 0$
  2. $8x - 9y = 0$
  3. $9x - 8y = 0$
  4. $9x + 8y = 0$

Part 1.

Shift the origin to the pole $(-6, 4)$ via $x \to x - 6$, $y \to y + 4$ in $4x^2 + 9y^2 = 144$:
$4(x-6)^2 + 9(y+4)^2 = 144$
$4x^2 - 48x + 144 + 9y^2 + 72y + 144 = 144$
$\boxed{4x^2 + 9y^2 - 48x + 72y + 144 = 0}$

Part 2.

For $\frac{x^2}{36} + \frac{y^2}{16} = 1$ ($a^2 = 36$, $b^2 = 16$), the diameter bisecting chords of slope $m$ has slope $-\frac{b^2}{a^2 m}$.
With $m = -\frac{1}{3}$:
$\text{slope} = -\frac{16}{36\left(-\frac{1}{3}\right)} = \frac{16}{12} = \frac{4}{3}$
So $y = \frac{4}{3}x$:
$\boxed{4x - 3y = 0}$

Part 3.

Using the same rule, slope of the bisecting diameter $= -\frac{b^2}{a^2 m}$ with $m = \frac{1}{2}$:
$\text{slope} = -\frac{16}{36\left(\frac{1}{2}\right)} = -\frac{16}{18} = -\frac{8}{9}$
So $y = -\frac{8}{9}x$:
$\boxed{8x + 9y = 0}$

Question Bank: t696

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

For the curve 36x^2 + 25y^2 - 900 = 0.

What is the equation of its diameter bisecting the chords having equal slope of -1/2?

  1. 72x - 25y = 0
  2. 36x + 25y = 0
  3. 72x + 25y = 0
  4. 72x - 50y = 0

What is the curvature of the curve at x = 3?

  1. -1.25
  2. -0.67
  3. - 0.31
  4. -0.86

What is the second eccentricity of the curve?

  1. 0.524
  2. 0.663
  3. 0.725
  4. 0.825

Part 1.

Rewrite: $\frac{x^2}{25} + \frac{y^2}{36} = 1$ ($a^2 = 25$ under $x^2$, $b^2 = 36$ under $y^2$). The diameter bisecting chords of slope $m$ has slope $-\frac{b^2}{a^2 m}$.
With $m = -\frac{1}{2}$:
$\text{slope} = -\frac{36}{25\left(-\frac{1}{2}\right)} = \frac{36}{12.5} = \frac{72}{25}$
So $y = \frac{72}{25}x$:
$\boxed{72x - 25y = 0}$

Part 3.

For $\frac{x^2}{25} + \frac{y^2}{36} = 1$, the major semi-axis is $a = 6$ and minor $b = 5$, with $c = \sqrt{a^2 - b^2} = \sqrt{11}$.
The second eccentricity uses the minor semi-axis:
$e' = \frac{c}{b} = \frac{\sqrt{11}}{5}$
$\boxed{e' = 0.663}$