Question

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

• A. $7a$
• B. $5a$
• C. $2a$
• D. $3a$

Answer 1

The correct option is B $5a$

Assume, the point $P$ is $(a{t}^2,2at)$ and the point $Q(\frac{a}{t^2},\frac{-2a}{t})$ of focal chord $PQ$.
The point of intersection of tangents at $P$ and $Q$ is $(-a,a(t-\frac{1}{t}))$ and the point lies on the line $y=2x+a.$ Hence, the intersection point satisfies the equation of line.

$a(t-\frac{1}{t})=-2a+a\Rightarrow t-\frac{1}{t}=-1\Rightarrow {(t-\frac{1}{t})}^2=1\therefore {(t+\frac{1}{t})}^2=5$
The length of the chord $PQ$ is calculated as,
Length of $PQ=a{(t+\frac{1}{t})}^2=5a$