Let P(at2, 2at), Q, R(ar2, 2ar) be three points on a parabola y2 = 4ax. If PQ is the focal chord and PK, QR are parallel where the co-ordinates of K is (2a, 0), then the value of r is

Question

Let P(at2, 2at), Q, R(ar2, 2ar) be three points on a parabola y2 = 4ax. If PQ is the focal chord and PK, QR are parallel where the co-ordinates of K is (2a, 0), then the value of r is (A) (t/1-t2) (B) (1-t2/t) (C) (t2 + 1/t) (D) (t2 - 1/t)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/8ed1c8fef3cfa7d4046fc614b37f5c26-.png" /> Here, coordinate of Q will be ((a/t2), (-2 a/t)): Slope of Q R=(2/r-(1)t) Slope of P K=(2 a t/a t2-2 a)=(2 t/t2-2) Since, Slope of Q R= Slope of P K ∴ (2/r-(1)t)=(2 t/t2-2) ⇒ r=(t2-1/t)

Q. Let $P (a t^{2}, 2 a t), Q, R (a r^{2}, 2 a r)$ be three points on a parabola $y^{2} = 4 a x$ . If $PQ$ is the focal chord and $P K, QR$ are parallel where the co-ordinates of $K$ is $(2 a, 0)$ , then the value of $r$ is

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

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

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

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

Q. Let P(at2,2at),Q,R(ar2,2ar) be three points on a parabola y2=4ax. If PQ is the focal chord and PK,QR are parallel where the co-ordinates of K is (2a,0), then the value of r is

1−t2t​

t1−t2​

tt2+1​

tt2−1​

Here, coordinate of Q will be (t2a​,t−2a​) : Slope of QR=r−t1​2​ Slope of PK=at2−2a2at​=t2−22t​ Since, Slope of QR= Slope of PK ∴r−t1​2​=t2−22t​ ⇒r=tt2−1​

Q. Let $P (a t^{2}, 2 a t), Q, R (a r^{2}, 2 a r)$ be three points on a parabola $y^{2} = 4 a x$ . If $PQ$ is the focal chord and $P K, QR$ are parallel where the co-ordinates of $K$ is $(2 a, 0)$ , then the value of $r$ is

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

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

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

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