Question

If $P({at}^2,2at)$ be one end of a focal chord of the parabola $y^2=4ax$, then the length of the chord is

• A. ${a(t-\frac{1}{t})}^2$
• B. $a(t-\frac{1}{t})$
• C. $a(t+\frac{1}{t})$
• D. ${a(t+\frac{1}{t})}^2$

Answer 1

The correct option is D ${a(t+\frac{1}{t})}^2$

If $P({at}^2,2at)$ be one end of a focal chord of the parabola $y^2=4ax$, then another end of chord will be $Q(\frac{a}{t^2},\frac{-2a}{t})$.
$\therefore$ Length of focal chord = PQ
$=\sqrt{{(\frac{a}{t^2}-{at}^2)}^2+{(-\frac{2a}{t}-2at)}^2}$
$=a\sqrt{{(\frac{1}{t}-t)}^2{(\frac{1}{t}+t)}^2+4{(\frac{1}{t}+t)}^2}$
$=a(\frac{1}{t}+t)\sqrt{{(\frac{1}{t}-t)}^2+4}$
$=a{(\frac{1}{t}+t)}^2$