Question

Statement-I : If $P_1{Q}_1$ and $P_2{Q}_2$ are two focal chords of the parabola $y^2=4ax$, 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 $P_1\left(a_1^2,2a_1\right)\&Q_1\left(\frac{a}{t_1^2},\frac{-2a}{t_1}\right)$
$P_2\left(a{t}_2^2,2a{t}_2\right)\&Q_2\left(\frac{a}{t_2^2},\frac{-2a}{t_2}\right)$
on $y^2=4ax$
equation of $P_1{P}_2$ :
$\left(t_1+t_2\right)y=2x+2t_1{t}_2$...(i)
equation of $Q_1{Q}_2$
$-\left(t_1+t_2\right)y=2x{t}_1{t}_2+2a$...(ii)
add (i) & (ii)
$x=-a$ which is directrix of $y^2=4ax$
Locus of point of intersection of tangent is directrix.
In case of parabola director circle is directrix