Question

Let chords of the circle $x^2+y^2=a^2$ touch the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$ Then their middle points lie on the curve

• A. $(x^2-y^2)=a^2{x}^2-b^2{y}^2$
• B. $(x^2+y^2{)}^2=a^2{x}^2-b^2{y}^2$
• C. $(x^2-y^2{)}^2=a^2{x}^2-b^2{y}^2$
• D. $(x^2+y^2)=a^2{x}^2-b^2{y}^2$

Answer 1

The correct option is B $(x^2+y^2{)}^2=a^2{x}^2-b^2{y}^2$

The equation of the chord of the circle $x^2+y^2=a^2$ whose middle point is $(\alpha ,\beta )$ is
$x\alpha +y\beta ={\alpha }^2+{\beta }^2\Rightarrow y=-\frac{\alpha }{\beta }x+\frac{{\alpha }^2+{\beta }^2}{\beta }$
Now the above equation is the tangent to the hyperbola
using the tangency condition
$m=-\frac{\alpha }{\beta },c^2=a^2{m}^2-b^2\Rightarrow {\left(\frac{{\alpha }^2+{\beta }^2}{\beta }\right)}^2=a^2\frac{{\alpha }^2}{{\beta }^2}-b^2$
$\Rightarrow ({\alpha }^2+{\beta }^2{)}^2=a^2{\alpha }^2-b^2{\beta }^2$
Hence, the locus of the middle point is
$(x^2+y^2{)}^2=a^2{x}^2-b^2{y}^2$