If s t = 1 , then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is [2014]

Question

Let $a, r, s, t$ be nonzero real numbers. Let $P (a t^{2}, 2 a t), Q, R (a r^{2}, 2 a r)$ and $S (a s^{2}, 2 a s)$ be distinct points on the parabola $y^{2} = 4 a x$ . Suppose that $P Q$ is the focal chord and lines QR and PK are parallel, where $K$ is the point $(2 a, 0)$ . [2014]

Q. If $s t = 1$ , then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

Smart Achievers · Accepted Answer

(2)

$Tangent at P is t y = x + a t^{2} . . . (i)$

$Normal at S is s x + y = 2 a s + a s^{3} . . . (i i)$

$y^{2} = 4 a x$

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

Now putting the value of $x$ from equation (i) in above equation, we get

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

$\Rightarrow 2 t^{3} y = a (t^{4} + 2 t^{2} + 1)$

$∴$ $y = \frac{a (t^{4} + 2 t^{2} + 1)}{2 t^{3}} = \frac{a {(t^{2} + 1)}^{2}}{2 t^{3}}$

Solution

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

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

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

4 $\frac{a {(t^{2} + 2)}^{2}}{t^{3}}$

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Solution

1 (t2+1)22t3

2 a(t2+1)22t3

3 a(t2+1)2t3

4 a(t2+2)2t3

Ans. (2) Tangent at P is ty=x+at2 ...(i) Normal at S is sx+y=2as+as3 ...(ii) Given st=1⇒s=1t ∴ xt+y=2at+at3⇒xt2+yt3=2at2+a Now putting the value of x from equation (i) in above equation, we get ⇒t2(ty-at2)+yt3=2at2+a ⇒2t3y=a(t4+2t2+1) ∴ y=a(t4+2t2+1)2t3=a(t2+1)22t3

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

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

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

4 $\frac{a {(t^{2} + 2)}^{2}}{t^{3}}$