Length of chord PQ is, Let be a focal chord of the parabola . The tangents to the parabola at and meet at a point lying on the line . [2013]

Question

Let $P Q$ be a focal chord of the parabola $y^{2} = 4 a x$ . The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y = 2 x + a, a > 0$ . [2013]

Q. Length of chord PQ is

Smart Achievers · Accepted Answer

(2)

$P Q = \sqrt{{(a t^{2} - \frac{a}{t^{2}})}^{2} + {(2 a t + \frac{2 a}{t})}^{2}}$

$= a \sqrt{{(t + \frac{1}{t})}^{2} {(t - \frac{1}{t})}^{2} + 4 {(t + \frac{1}{t})}^{2}}$

$= a (t + \frac{1}{t}) \sqrt{{(t - \frac{1}{t})}^{2} + 4}$

$= a {(t + \frac{1}{t})}^{2} = 5 a$

Solution

Q.
Let $P Q$ be a focal chord of the parabola $y^{2} = 4 a x$ . The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y = 2 x + a, a > 0$ . [2013]

Q. Length of chord PQ is

1 $7 a$

2 $5 a$

3 $2 a$

4 $3 a$

Ans.
(2)

$P Q = \sqrt{{(a t^{2} - \frac{a}{t^{2}})}^{2} + {(2 a t + \frac{2 a}{t})}^{2}}$

$= a \sqrt{{(t + \frac{1}{t})}^{2} {(t - \frac{1}{t})}^{2} + 4 {(t + \frac{1}{t})}^{2}}$

$= a (t + \frac{1}{t}) \sqrt{{(t - \frac{1}{t})}^{2} + 4}$

$= a {(t + \frac{1}{t})}^{2} = 5 a$

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Solution

Q. Let PQ be a focal chord of the parabola y2=4ax. The tangents to the parabola at P and Q meet at a point lying on the line y=2x+a, a>0. [2013] Q. Length of chord PQ is

1 7a

2 5a

3 2a

4 3a

Ans. (2) PQ=(at2-at2)2+(2at+2at)2 =a(t+1t)2(t-1t)2+4(t+1t)2 =a(t+1t)(t-1t)2+4 =a(t+1t)2=5a