Let $\mathrm{a}, \mathrm{r}, \mathrm{s}, \mathrm{t}$ be nonzero real numbers. Let $\mathrm{P}\left(\mathrm{at}^2, 2 \mathrm{at}\right), \mathrm{Q}, \mathrm{R}\left(\mathrm{ar}^2, 2 \mathrm{ar}\right)$ and (as ${ }^2, 2 \mathrm{as}$ ) be distinct points on the parabola $\mathrm{y}^2=4 \mathrm{ax}$. 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
Select the correct option:
A
$\frac{\left(t^2+1\right)^2}{2 t^3}$
B
$\frac{a\left(t^2+1\right)^2}{2 t^3}$
C
$\frac{a\left(t^2+1\right)^2}{t^3}$
D
$\frac{a\left(t^2+2\right)^2}{t^3}$
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
Tangent at P is $\mathrm{ty}=\mathrm{x}+\mathrm{at}^2$
Normal at $S$ is $y+s x=2 a s+a s^2$
$\begin{aligned} & t y+x=2 a+\frac{a}{t^2} \\ & t y=2 a+\frac{a}{t^2}-t y+a t^2 \\ & 2 t^3 y=a t^4+2 a t^2+a \\ & y=\frac{a\left(t^2+1\right)^2}{2 t^3}\end{aligned}$
Hello 👋 Welcome to Competishun – India’s most trusted platform for JEE & NEET preparation. Need help with JEE / NEET courses, fees, batches, test series or free study material? Chat with us now 👇
