Locus of hyperbole chords midpoints

Question

Find the locus of midpoints of all chords of hyperbole $b^2x^2-a^2y^2=a^2b^2$ that pass through point $A(a,0)$.

How do I approach this?

Answer 1

HINT:

The equation of the any straight line passing through $A(a,0)$ is $$\dfrac{y-0}{x-a}=m\iff y=mx-ma$$ where $m$ is the gradient

Put this value of $y$ in $$b^2x^2-a^2y^2=a^2b^2$$ to form a Quadratic Equation in $x$ whose roots represent the abscissa of the intersection.

So, if the midpoint of the intersection if $P(h,k)$

$$h=\dfrac{x_1+x_2}2$$

Similarly, $k=\dfrac{y_1+y_2}2,$ where $y_i=mx_i-ma;i=1,2$

Now eliminate $m$