Paragraph: Let $a, r, s, t$ be nonzero 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…

Question

Paragraph: Let $a, r, s, t$ be nonzero 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 $(2 a, 0)$. Question: The value of $r$ is

Aicharya · Accepted Answer

Slope (QR) = slope (PK) 2 a t - 0 a t 2 - 2 a = - 2 a t - 2 a r a t 2 - a r 2 &#160; &#8658; &#160; &#160; t t 2 - 2 = - 1 t + r 1 t 2 - r 2 &#160; &#160; &#8658; &#160; &#160; &#160; r = t 2 - 1 t

Paragraph: Let $a, r, s, t$ be nonzero real numbers. Let $P\left(a t^{2}, 2 a t\right), Q, R\left(a r^{2}, 2…

Solution