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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.

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 $m$:

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.

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 $x - y = 4$.

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 $x - y = 4$.