Question

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

• A. The triangle $PFQ$ is a right-angled triangle
• B. The triangle $QPQ '$ is a right-angle triangle
• C. The distance between $P$ and $F$ is $5 \sqrt{2}$
• D. $F$ lies on the line joining $Q$ and $Q'$

Answer 1

The Correct Option is A, B, D

Step 1: Understanding the Parabola and Given Points
We are given the parabola $y^2 = 8x$. The equation of the parabola represents a standard parabola that opens to the right with the vertex at the origin $(0, 0)$.
The focus of the parabola, denoted by $F$, is at $(2, 0)$, since the general formula for a parabola of the form $y^2 = 4ax$ gives the focus at $(a, 0)$. Here, $a = 2$, so the focus is at $F(2, 0)$.
We are also given the point $P = (-2, 4)$ and two distinct points $Q$ and $Q'$ on the parabola such that the lines $PQ$ and $PQ'$ are tangents to the parabola at $Q$ and $Q'$. We need to determine which of the following statements is true:

• (A) The triangle $PFQ$ is a right-angled triangle
• (B) The triangle $QPQ'$ is a right-angled triangle
• (D) $F$ lies on the line joining $Q$ and $Q'$

Step 2: Statement (A) - The triangle $PFQ$ is a right-angled triangle
To determine if the triangle $PFQ$ is a right-angled triangle, we need to check if the angle $\angle PFQ$ is $90^\circ$. We know that the tangent at any point on the parabola makes an angle of $90^\circ$ with the line joining the point of tangency and the focus. Hence, since $PQ$ is tangent to the parabola at $Q$, and the focus is at $F$, the angle $\angle PFQ$ is indeed a right angle.
Therefore, statement (A) is TRUE. 
Step 3: Statement (B) - The triangle $QPQ'$ is a right-angled triangle
For triangle $QPQ'$ to be right-angled, one of its angles must be $90^\circ$. The points $Q$ and $Q'$ are distinct points on the parabola, and the lines $PQ$ and $PQ'$ are tangents to the parabola at these points. Since the tangents at two distinct points on the parabola are perpendicular to the line joining those points with the focus, the angle between $PQ$ and $PQ'$ is $90^\circ$. This makes triangle $QPQ'$ a right-angled triangle.
Therefore, statement (B) is TRUE. 
Step 4: Statement (D) - $F$ lies on the line joining $Q$ and $Q'$
This statement is related to a well-known property of parabolas: the two tangents drawn from any external point to a parabola always meet at the same angle at the focus. Since $Q$ and $Q'$ are points of tangency, the line joining them will pass through the focus $F$. This is because the tangents at $Q$ and $Q'$ must be symmetric with respect to the line joining $Q$ and $Q'$, and the focus lies on this line.
Therefore, statement (D) is TRUE. 
Step 5: Conclusion
The correct statements are (A), (B), and (D). Thus, the answer is:

TRUE for all statements: (A), (B), (D).