Let tangents $PQ$ and $PR$ are drawn from the point $P(-2, 4)$ to the parabola $y^2 = 4x$. If $S$ is the focus of the parabola $y^2 = 4x$, then the value (in units) of $RS + SQ$ is equal to
Step-by-Step Solution
Key Concept: Use parametric representation of points on a parabola and algebraic identities like $t_1^2 + t_2^2 = (t_1 + t_2)^2 - 2t_1t_2$ to simplify sums of distances.
Given $Q = (t_1^2, 2t_1)$ and $R = (t_2^2, 2t_2)$, with $P = (t_1 t_2, t_1 + t_2) = (-2, 4)$. Thus $t_1 t_2 = -2$ and $t_1 + t_2 = 4$. Now, $QS = 1 + t_1^2$ and $RS = 1 + t_2^2$. We have $QS + RS = 2 + t_1^2 + t_2^2 = 2 + (t_1 + t_2)^2 - 2t_1t_2 = 2 + (4)^2 - 2(-2) = 2 + 16 + 4 = 22$ units.
Correct Answer: 22