Let E denote the parabola y 2 = 8 x . Let P = ( - 2 , 4 ) and let Q and Q be two distinct points on E such that the lines P Q and P Q are tangents to E . Let F be the focus of E . Then which of the following statements is (are) TRUE?

Question

Let $E$ denote the parabola $y^{2} = 8 x$ . Let $P = (- 2, 4)$ and let $Q$ and $Q'$ be two distinct points on $E$ such that the lines $P Q$ and $P Q'$ are tangents to $E$ . Let $F$ be the focus of $E$ . Then which of the following statements is (are) TRUE? [2021]

Smart Achievers · Accepted Answer

(1, 2, 4)

Given that

$E : y^{2} = 8 x,$

$P = (- 2, 4)$

Point $P (- 2, 4)$ lies on directrix of parabola.

So, $∠ Q P Q' = \frac{π}{2}$ and chord $Q Q'$ is a focal chord and segment PQ subtends right angle at the focus. So, $∠ P F Q = \frac{π}{2}$

Slope of $Q Q' = \frac{2}{t_{1} + t_{2}} = 1$

Slope of $P F = - 1$ $∴$ $Q Q' ⊥ P F$

$P F = 4 \sqrt{2}$

Solution

Q.
Let $E$ denote the parabola $y^{2} = 8 x$ . Let $P = (- 2, 4)$ and let $Q$ and $Q'$ be two distinct points on $E$ such that the lines $P Q$ and $P Q'$ are tangents to $E$ . Let $F$ be the focus of $E$ . Then which of the following statements is (are) TRUE [2021]

1 The triangle $P F Q$ is a right-angled triangle

2 The triangle $Q P Q'$ is a right-angled triangle

3 The distance between P and F is $5 \sqrt{2}$

4 F lies on the line joining $Q$ and $Q'$

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Solution

Q. Let E denote the parabola y2=8x. Let P=(-2,4) and let Q and Q' be two distinct points on E such that the lines PQ and PQ' are tangents to E. Let F be the focus of E. Then which of the following statements is (are) TRUE [2021]

1 The triangle PFQ is a right-angled triangle

2 The triangle QPQ' is a right-angled triangle

3 The distance between P and F is 52

4 F lies on the line joining Q and Q'