jee-advanced 2022 Q13

jee-advanced · India · paper1 4 marks Circles Tangent Lines and Tangent Lengths
Consider the parabola $y ^ { 2 } = 4 x$. Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P = ( - 2,1 )$ meet the parabola at $P _ { 1 }$ and $P _ { 2 }$. Let $Q _ { 1 }$ and $Q _ { 2 }$ be points on the lines $S P _ { 1 }$ and $S P _ { 2 }$ respectively such that $P Q _ { 1 }$ is perpendicular to $S P _ { 1 }$ and $P Q _ { 2 }$ is perpendicular to $S P _ { 2 }$. Then, which of the following is/are TRUE?
(A) $\quad S Q _ { 1 } = 2$
(B) $\quad Q _ { 1 } Q _ { 2 } = \frac { 3 \sqrt { 10 } } { 5 }$
(C) $\quad P Q _ { 1 } = 3$
(D) $\quad S Q _ { 2 } = 1$
Consider the parabola $y ^ { 2 } = 4 x$. Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P = ( - 2,1 )$ meet the parabola at $P _ { 1 }$ and $P _ { 2 }$. Let $Q _ { 1 }$ and $Q _ { 2 }$ be points on the lines $S P _ { 1 }$ and $S P _ { 2 }$ respectively such that $P Q _ { 1 }$ is perpendicular to $S P _ { 1 }$ and $P Q _ { 2 }$ is perpendicular to $S P _ { 2 }$. Then, which of the following is/are TRUE?\\
(A) $\quad S Q _ { 1 } = 2$\\
(B) $\quad Q _ { 1 } Q _ { 2 } = \frac { 3 \sqrt { 10 } } { 5 }$\\
(C) $\quad P Q _ { 1 } = 3$\\
(D) $\quad S Q _ { 2 } = 1$