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

Question

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 (A) -(1/t) (B) (t2+1/t) (C) (1/t) (D) (t2-1/t)

Tardigrade Admin · Accepted Answer

Slope ( QR )= Slope ( PK ) (2 at -0/ at 2-2 a )=(-(2 a / t )-2 ar /( a ) t 2- ar 2) ⇒ ( t / t 2-2)=-(((1/ t )+ r /(1) t 2- r 2)) ⇒ r =( t 2-1/ t )

$- \frac{1}{t}$

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

$\frac{1}{t}$

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

Slope $(QR) =$ Slope $(P K)$
$\frac{2 a t - 0}{a t ^{2} - 2 a} = \frac{- \frac{2 a}{t} - 2 a r}{\frac{a}{t ^{2}} - a r ^{2}}$
$\Rightarrow \frac{t}{t ^{2} - 2} = - (\frac{\frac{1}{t} + r}{\frac{1}{t ^{2}} - r ^{2}})$
$\Rightarrow r = \frac{t ^{2} - 1}{t}$

Q. Let a,r,s,t be non-zero 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). The value of r is

−t1​

tt2+1​

t1​

tt2−1​

Slope (QR)= Slope (PK) at2−2a2at−0​=t2a​−ar2−t2a​−2ar​ ⇒t2−2t​=−(t21​−r2t1​+r​) ⇒r=tt2−1​

$- \frac{1}{t}$

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

$\frac{1}{t}$

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

Slope $(QR) =$ Slope $(P K)$
$\frac{2 a t - 0}{a t ^{2} - 2 a} = \frac{- \frac{2 a}{t} - 2 a r}{\frac{a}{t ^{2}} - a r ^{2}}$
$\Rightarrow \frac{t}{t ^{2} - 2} = - (\frac{\frac{1}{t} + r}{\frac{1}{t ^{2}} - r ^{2}})$
$\Rightarrow r = \frac{t ^{2} - 1}{t}$