The value of r is, 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 PQ is the focal chord and lines QR and PK are

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 PQ is the focal chord and lines QR and PK are parallel, where $K$ is the point $(2 a, 0)$ . [2014]

Q. The value of $r$ is

Smart Achievers · Accepted Answer

(4)

$∵$ $P Q is a focal chord, Q (\frac{a}{t^{2}}, \frac{- 2 a}{t})$

$Also Q R ∥ P K,$ $∴$ $m_{Q R} = m_{P K}$

$\Rightarrow \frac{\frac{- 2 a}{t} - 2 a r}{\frac{a}{t^{2}} - a r^{2}} = \frac{0 - 2 a t}{2 a - a t^{2}}$

$\Rightarrow \frac{- 2 a (\frac{1}{t} + r)}{a (\frac{1}{t} + r) (\frac{1}{t} - r)} = \frac{- 2 a t}{a (2 - t^{2})}$

$\Rightarrow 2 - t^{2} = t (\frac{1}{t} - r) [∵ r \neq - \frac{1}{t} otherwise Q will coincide with R]$

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

Solution

1 $- \frac{1}{t}$

2 $\frac{t^{2} + 1}{t}$

3 $\frac{1}{t}$

4 $\frac{t^{2} - 1}{t}$

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Solution

Q. Let a,r,s,t be nonzero real numbers. Let P(at2,2at), Q, R(ar2,2ar) and S(as2,2as) be distinct points 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). [2014] Q. The value of r is

1 -1t

2 t2+1t

3 1t

4 t2-1t

Ans. (4) ∵ PQ is a focal chord, Q(at2,-2at) Also QR∥PK, ∴ mQR=mPK ⇒-2at-2arat2-ar2=0-2at2a-at2 ⇒-2a(1t+r)a(1t+r)(1t-r)=-2ata(2-t2) ⇒2-t2=t(1t-r) [∵r≠-1t otherwise Q will coincide with R] ⇒2-t2=1-tr⇒r=t2-1t

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1 $- \frac{1}{t}$

2 $\frac{t^{2} + 1}{t}$

3 $\frac{1}{t}$

4 $\frac{t^{2} - 1}{t}$