1. Tardigrade
  2. Question
  3. Mathematics
  4. Let a, r, s, t be non-zero real numbers. Let P(a t2, 2 a t), Q, R(a r2, 2 a r) and S(a s2, 2 a s) be distinct points on the parabola y2=4 a x. Suppose that P Q is the focal chord and lines Q R and P K are parallel, where K is the point (2 a, 0). The value of r is

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Q. Let $a, r, s, t$ be non-zero real numbers. Let $P\left(a t^{2}, 2 a t\right), Q, R\left(a r^{2}, 2 a r\right)$ and $S\left(a s^{2}, 2 a s\right)$ be distinct points on the parabola $y^{2}=4 a x$. Suppose that $P Q$ is the focal chord and lines $Q R$ and $P K$ are parallel, where $K$ is the point $(2 a, 0)$.
The value of $r$ is

JEE AdvancedJEE Advanced 2014

Solution:

Slope $( QR )=$ Slope $( PK )$
$\frac{2 at -0}{ at ^{2}-2 a }=\frac{-\frac{2 a }{ t }-2 ar }{\frac{ a }{ t ^{2}}- ar ^{2}}$
$\Rightarrow \frac{ t }{ t ^{2}-2}=-\left(\frac{\frac{1}{ t }+ r }{\frac{1}{ t ^{2}}- r ^{2}}\right) $
$\Rightarrow r =\frac{ t ^{2}-1}{ t }$