Discussion Forum

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)

The value of r is 

Answer:

Explanation:

 Plan 

 (i)  P( at², 2at) is one end point of focal chord of parabola y²=4ax,

 then other end point is   $(\frac{a}{t^{2}},-\frac{2a}{t})$

 (ii)  Slope of line joining two points  (x₁,y₁) and (x₂,y₂) is given by  $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

If PQ is foacl  chord, then coordinates of Q will be $(\frac{a}{t^{2}},-\frac{2a}{t})$

 Now, slope of QR= slope of PK

  $\frac{2ar+2a/t}{ar^{2}-a/t^{2}}=\frac{2at}{at^{2}-2a}$

 $\Rightarrow$    $\frac{r+1/t}{r^{2}-1/t^{2}}=\frac{t}{t^{2}-2}$

 $\Rightarrow$    $\frac{1}{r-1/t}=\frac{t}{t^{2}-2}$

$\Rightarrow$     $r-\frac{1}{t}=\frac{t^{2}-2}{t}=t-\frac{2}{t}$

  $\Rightarrow$     $r=  t-\frac{1}{t}=\frac{t^{2}-1}{t}$