Discussion Forum

 Suppose that the foci of the ellipse   $\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$  are (f₁,0) and (f₂, o) where f₁>0 and f₂<0 . Let P₁   and P₂ be two parabolas with a common vertex at (0,0) with foci at (f₁,0) and (2f₂,0)  respectively  .Let T₁ be a tangent to P₁ which passes through (2 f₂,o) and T₂ be  a tangent to P₂. P2 which passes through (f₁,0) . If m₁ is the slop of T₁  and m₂ is the slope of T₂, then the value of  $\left( \frac{1}{m_1^2}+m_2^2\right)$  is

Answer:

Explanation:

Tangent to P₁ passes through

   (2f₂,0)i.e, (-4,0)

  $\therefore  T_{1}:y=m_{1}x+\frac{2}{m_{1}}$

 $\Rightarrow$              $0=-4m_{1}+\frac{2}{m_{1}}$

$\Rightarrow$        $m_{1}^{2}=1/2$     ..........(i)

Also, tangent to  P₂   passes through (f₁,0) i.e, (2,0)

     $\Rightarrow$                $T_{2}:y=m_{2}x+\frac{(-4)}{m_{2}}$

$\Rightarrow$                     $0=2m_{2}-\frac{4}{m_{2}}$

$\Rightarrow$               $m_2^2=2$

  $\therefore$    $\frac{1}{m_1^2}+m_2^2=2+2=4$