Question

If $P(a{t}^2,2at)$ 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 $P(a{t}^2,2at)$ 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 $=PQ$

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

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

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

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