If chord P Q subtends an angle at the vertex of y 2 = 4 a x , then tan , 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]

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. If chord $P Q$ subtends an angle $θ$ at the vertex of $y^{2} = 4 a x$ , then $\tan θ =$

Smart Achievers · Accepted Answer

(4)

$Since P Q is the focal chord of y^{2} = 4 a x$

$∴$ $Coordinates of P and Q can be taken as$

$P (a t^{2}, 2 a t) and Q (\frac{a}{t^{2}}, \frac{- 2 a}{t})$

$Equation of tangents at P and Q are$

$y = \frac{x}{t} + a t and y = - x t - \frac{a}{t},$

$which intersect each other at R (- a, a (t - \frac{1}{t}))$

$As R lies on the y = 2 x + a, a > 0$

$∴$ $a (t - \frac{1}{t}) = - 2 a + a \Rightarrow t - \frac{1}{t} = - 1 \Rightarrow t + \frac{1}{t} = \sqrt{5}$

$Now, m_{O P} = \frac{2}{t} and m_{O Q} = - 2 t$

$∴$ $\tan θ = \frac{\frac{2}{t} + 2 t}{1 - 4} = \frac{2 (t + \frac{1}{t})}{- 3} = \frac{2 \sqrt{5}}{- 3}$

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. If chord $P Q$ subtends an angle $θ$ at the vertex of $y^{2} = 4 a x$ , then $\tan θ =$

1 $\frac{2}{3} \sqrt{7}$

2 $\frac{- 2}{3} \sqrt{7}$

3 $\frac{2}{3} \sqrt{5}$

4 $\frac{- 2}{3} \sqrt{5}$

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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. If chord PQ subtends an angle θ at the vertex of y2=4ax, then tanθ=

1 237

2 -237

3 235