Question

Give the ellipse 9x² + 16y2 = 144, find the equation of the diameter which bisects the chord with slope 1/2​

Answer 1

The equation of the diameter which bisects the chord with slope 1/2 is

$y=-\frac{9}{8} x$

The general form of the equation of given ellipse is  $ax^2+by^2=c$ and equation of chord will be $y=mx+n$ for some m and n.

Now in order to find values of x that intercept the ellipse it must fulfill the equation as such;

$ax^2+b(mx+n)^2=c$

$(a+bm^2)x^2+2bmnx+bn^2 - c=0$

What is the definition of an ellipse?

It is a plane section curve whose sum from two fixed points will be a constant value.

Quadratic roots will give the mid value of x and value of y will be ;

$X=\frac{-b}{2a} =\frac{-bmn}{a+bm^2}$

$Y=n-\frac{bm^2n}{a+bm^2} =\frac{an}{a+bm^2}$

Thus the final equation of required diameter will be :

$y=\frac{Y}{X} x=-\frac{a}{bm} x$

And in this case the equation of the diameter which bisects the chord with slope 1/2 is,

$y=-\frac{9}{8} x$

Learn more from here quadratic equations,

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