Finding Locus of mid point of portion of tangents

Question

Finding locus of middle points of tangents to the circle: $ x^2 + y^2 = a ^2$ terminated by the coordinate axis.

I am not able to figure out what the question wanna say... any help is appreciated.

Answer 1

Let $(h,k)$ be any point on the circle $x^2+y^2=a^2$. The tangent to the circle at the point $(h,k)$ is $hx+ky=a^2$ The coordinate of the point where the tangent meets the $x$-axis and $y$-axis is $A:(\frac{a^2}{h},0)$ and $B:(o,\frac{a^2}{k})$ resp. The midpoint of the line joining the line segment $AB$ is $(\alpha, \beta ):(\frac{a^2}{2h},\frac{a^2}{2k})$ . We need to find locus of $(\alpha, \beta )$

Now,

$\alpha= \frac{a^2}{2h} \implies h=\frac{a^2}{2\alpha}---(1)$

$\beta= \frac{a^2}{2k} \implies k=\frac{a^2}{2\beta}---(2)$

But, $(h,k)$ lies on the circle, therefore, $h^2+k^2=a^2$

Substituting values of $(1)$ and $(2)$ in $(3)$ we get

$\frac{a^4}{4\alpha ^2}+\frac{a^4}{4\beta ^2}=a^2$

Changing $(\alpha, \beta)$ to co-ordinates $(x,y)$ we get the equation of the desired locus: $\frac{a^4}{4 x ^2}+\frac{a^4}{4 y ^2}=a^2$.