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

Answer 1

Answer: (D)

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 $Q \left(\frac{a}{t^{2}}, \frac{-2a}{t}\right)$.

$\therefore $ Length of focal chord $= PQ$

$=\sqrt{\left(\frac{a}{t^{2}} -at^{2}\right)^{2} + \left(-\frac{2a}{t} -2at\right)^{2}} $

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

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

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