Question

Statement-I : If $P _1 Q _1$ and $P _2 Q _2$ are two focal chords of the parabola $y ^2=4 ax$, then the locus of point of intersection of chords $P _1 P _2$ and $Q _1 Q _2$ is directrix of the parabola. Here $P _1 P _2$ and $Q _1 Q _2$ are not parallel.
Because
Statement-II : The locus of point of intersection of perpendicular tangents of parabola is directrix of parabola.

• A. Statement-I is true, Statement-II is true ; Statement-II is correct explanation for Statement-I.
• B. Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for Statement-I.
• C. Statement-I is true, Statement-II is false.
• D. Statement-I is false, Statement-II is true

Answer 1

Answer: (B)

Let $P_1\left(a_1^2, 2 a_1\right) \& Q_1\left(\frac{a}{t_1^2}, \frac{-2 a}{t_1}\right)$
$P _2\left( at _2^2, 2 at _2\right) \& Q _2\left(\frac{ a }{ t _2^2}, \frac{-2 a }{ t _2}\right)$
on $y^2=4 a x$
equation of $P _1 P _2$ :
$\left(t_1+t_2\right) y=2 x+2 t_1 t_2$...(i)
equation of $Q _1 Q _2$
$-\left(t_1+t_2\right) y=2 x t_1 t_2+2 a$...(ii)
add (i) & (ii)
$x=-a$ which is directrix of $y^2=4 a x$
Locus of point of intersection of tangent is directrix.
In case of parabola director circle is directrix