Discussion Forum



1)

Let a,r, s, t be non-zero real numbers. Let  P( at2,2at) , Q , R (ar2, 2ar) and S( as2, 2as) be distinct point on the parabola y2=4ax . Suppose  that PQ is the  focal  chord and  lines QR and PK are parallel , where K is the point (2a,0)

 If st=1 , then the tangent at P  and the normal at S to the parabola meet at a point  whose ordinate is 


A) $\frac{(t^{2}+1)^{2}}{2t^{3}}$

B) $\frac{a(t^{2}+1)^{2}}{2t^{3}}$

C) $\frac{a(t^{2}+1)^{2}}{t^{3}}$

D) $\frac{a(t^{2}+2)^{2}}{t^{3}}$

Answer:

Option B

Explanation:

Plan  Equation  of tangent and normal at (at2, 2at) are given by ty= x+at2  and y+t x= 2at+at3 , respectively

  Tangent at P :ty= x+at2

  or      $y= \frac{x}{t}+at$

 Normal at S:

  $y+ \frac{x}{t}=\frac{2a}{t}+\frac{a}{t^{3}}$

 Solving   $2y=at+\frac{2a}{t}+\frac{a}{t^{3}}$

 $\Rightarrow$     $ y=\frac{a(t^{2}+1)^{2}}{2t^{3}}$