Consider the parabola $y^{2} = 4x$. 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 $SP_1$ and $SP_2$ respectively such that $PQ_1$ is perpendicular to $SP_1$ and $PQ_2$ is perpendicular to $SP_2$. Then, which of the following is/are TRUE?

Consider the parabola y² = 4x. 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 P₁ and P₂. Let Q₁ and Q₂ be points on the lines SP₁ and SP₂ respectively such that PQ₁ is perpendicular to SP₁ and PQ₂ is perpendicular to SP₂. Then, which of the following is/are TRUE? $\text{Given: parabola } y^2 = 4x,\ \text{focus } S$ $P = (-2,\ 1)$ $\text{Tangents from } P \text{ touch at } P_1,\ P_2$ $PQ_1 \perp SP_1,\quad PQ_2 \perp SP_2$ $\text{Check which options are true}$ $y^2 = 4x \Rightarrow a = 1,\quad S = (1,0)$ $1.\ \text{slopes of tangents from } P$ $2.\ \text{contact points } P_1,\ P_2$ $3.\ \text{feet } Q_1,\ Q_2 \Rightarrow \text{check options}$ $y = mx + \frac{1}{m}$ $1 = -2m + \frac{1}{m} \Rightarrow 2m^2 + m - 1 = 0$ $(2m-1)(m+1) = 0 \Rightarrow m = \frac{1}{2},\ -1$ $y = \frac{x}{2} + 2,\qquad y = -x - 1$ $\text{point } (at^2, 2at);\ \text{tangent slope} = \frac{1}{t} \Rightarrow t = \frac{1}{m}$ $m=\frac{1}{2} \Rightarrow P_1(4,4);\quad m=-1 \Rightarrow P_2(1,-2)$ $SP_2:\ x = 1 \Rightarrow Q_2 = (1,1)$ $SQ_2 = 1 \Rightarrow \text{option (4) TRUE}$ $SP_1:\ (1+3\lambda,\ 4\lambda)$ $(Q_1-P)\cdot(3,4)=0 \Rightarrow 5 + 25\lambda = 0 \Rightarrow \lambda = -\frac{1}{5}$ $Q_1 = \left(\frac{2}{5},\ -\frac{4}{5}\right)$ $PQ_1 = \sqrt{\left(\frac{12}{5}\right)^2 + \left(\frac{9}{5}\right)^2} = \frac{15}{5} = 3$ $\Rightarrow \text{option (3) TRUE}$ $SQ_1 = \sqrt{\frac{9+16}{25}} = 1 \neq 2$ $\Rightarrow \text{option (1) FALSE}$ $Q_1Q_2 = \frac{\sqrt{9+81}}{5} = \frac{3\sqrt{10}}{5}$ $\Rightarrow \text{option (2) TRUE}$ $\text{True: (2)},\ (3),\ (4)$ $\boxed{\text{Answer: (2), (3), (4)}}$