Discussion Forum

Let P  be a point on the circle S with both coordinates being positive. Let the tangent to S  at P intersect the coordinate  axes at the points  M and N.  Then the mid -point of the line segment MN must lie on the curve

Answer:

Explanation:

We have, x² + y² =4

 Let $P(2\cos\theta,2\sin\theta)$  be a point on a circle.

  Tangent at P is  $2\cos\theta x+2\sin\theta y=4$

  = $x\cos\theta +y\sin\theta =2$

   $\therefore$     The coordinates at  $M(\frac{2}{\cos\theta},0)$ and $N(0, \frac{2}{\sin\theta})$

    Let (h,k) is mid-point of MN

  $\therefore$  $h= \frac{1}{\cos\theta}$   and   $k= \frac{1}{\sin\theta}$

 $\Rightarrow$ $\cos\theta =\frac{1}{h}$  and         

 $\sin\theta =\frac{1}{k}$

$\Rightarrow$        $\cos^{2}\theta +\sin^{2}\theta= \frac{1}{h^{2}}+\frac{1}{k^{2}}$

  $\Rightarrow$   $1= \frac{h^{2}+k^{2}}{h^{2}.k^{2}}$

$\Rightarrow$      $h^{2}+k^{2}= h^{2}.k^{2}$

  $\therefore$   Mid-point of MN lie on the curve

  $x^{2}+y^{2}=x^{2}y^{2}$