Let E denote the parabola y² = 8x. Let P = (-2, 4), and let Q and Q' be two distinct points on E suc

Q: Wait, why do we reject the negative root here? Or why can't we just use the slope-point form to find the tangents?

You absolutely could use the slope-point form, but in JEE Advanced, time is your most precious resource. The beauty of conic sections lies in their elegant properties. By recognizing that $P$ lies on the directrix, you bypass complex algebraic derivations entirely. Always look for the geometric shortcut first!

Q: How do I know $P$ is on the directrix without plotting?

Great question! The standard form of a parabola is $y^2 = 4ax$. Here, $4a = 8$, so $a = 2$. The directrix is defined by the line $x = -a$, which is $x = -2$. Since the x-coordinate of $P$ is $-2$, it satisfies the equation of the directrix perfectly. It's a classic setup to test if you've internalized the definitions.

Q: Is the property about the angle at the focus always true for any point on the directrix?

Yes, it is! The tangent segment intercepted between the point of contact and the directrix always subtends a right angle at the focus. This is a powerful, standard property of parabolas that simplifies many complex problems. Keep this in your toolkit!

Question

Let $E$ denote the parabola $y^2 = 8x$. 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?

Options:

(A) The triangle $PFQ$ is a right-angled triangle

(B) The triangle $QPQ'$ is a right-angled triangle

(C) The distance between $P$ and $F$ is $5\sqrt{2}$

(D) $F$ lies on the line joining $Q$ and $Q'$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The journey begins by decoding the DNA of the parabola given by the equation $y^2 = 8x$. The standard form of a parabola is $y^2 = 4ax$.

By comparing the given equation with the standard form, we find $4a = 8$, which yields $a = 2$. This value is the fundamental parameter of our curve.

The focus $F$ is located at $(a, 0)$, which is $(2, 0)$. The directrix is defined by the line $x = -a$, or $x = -2$.

### The Directrix Revelation

Consider the point $P(-2, 4)$. Notice that its $x$-coordinate is $-2$, meaning it lies exactly on the directrix.

In the context of JEE geometry, when a point lies on the directrix, we must recall the properties of tangents. A fundamental property of the parabola is that tangents drawn from any point on the directrix are always perpendicular to each other.

Therefore, $\angle QPQ' = 90^\circ$. This confirms that $	riangle QPQ'$ is a right-angled triangle, making **Option B correct**.

### The Chord of Contact

Next, we examine the line $QQ'$, which represents the chord of contact. There is a beautiful theorem stating that the chord of contact drawn from any point on the directrix always passes through the focus.

This implies that $QQ'$ is a focal chord. Consequently, the focus $F$ must lie on the line $QQ'$, which confirms that **Option D is correct**.

### The Distance Dilemma

To evaluate Option C, we calculate the distance $PF$. Given $P(-2, 4)$ and $F(2, 0)$, we apply the distance formula:

$$PF = \sqrt{(2 - (-2))^2 + (0 - 4)^2}$$

$$PF = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$$

Since Option C claims the distance is $5\sqrt{2}$, it is mathematically false.

### The Final Elegance

Finally, we analyze Option A. We must determine if $	riangle PFQ$ is right-angled.

A powerful property of parabolas states that the tangent segment intercepted between the point of contact and the directrix subtends a right angle at the focus. Therefore, $\angle PFQ = 90^\circ$.

This confirms that $	riangle PFQ$ is a right-angled triangle, meaning **Option A is correct**.

We have navigated the geometry and verified the properties. The final conclusion is that **Options A, B, and D are correct**.

Select Answer:

Visualized Solution

Parabola Parameters

Focus and Directrix

Position of Point $P$

Tangents from Directrix

Checking Option B

Chord of Contact

Checking Option D

Distance $P F$ Setup

Checking Option C

Angle at Focus Setup

Checking Option A

Final Conclusion

The Sigma Insight: Equation of Tangent and Normal

Analyzing the Setup

The Directrix Revelation

The Chord of Contact

The Distance Dilemma

The Final Elegance

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