Let P : y² = 4ax, a 0 be a parabola with focus S. Let the tangents to the parabola P make an angle

Q: Wait, why do we use the absolute value in the angle formula $\tan(\theta) = \left| \frac{m - m_1}{1 + m \cdot m_1} \right|$?

Great question! The absolute value is necessary because the angle between two lines is defined as the acute angle. Since the formula $\frac{m - m_1}{1 + m \cdot m_1}$ can yield a negative value depending on which line you subtract from the other, the absolute value ensures we get the positive tangent of the acute angle $\theta = \frac{\pi}{4}$.

Q: Why does the product of the slopes being $-1$ matter so much?

In coordinate geometry, the product of the slopes of two lines being $-1$ is the definitive condition for perpendicularity. This is a massive shortcut! It tells us immediately that the two tangents are perpendicular, which unlocks a powerful geometric property of parabolas that saves us from doing tedious algebraic calculations.

Q: Is it always true that the chord of contact from the directrix passes through the focus?

Yes, it is a fundamental theorem of parabolas! Any point on the directrix of a parabola has a chord of contact that is a focal chord. This means the chord will always pass through the focus $S(a, 0)$. This is why the collinearity of $A, B,$ and $S$ is independent of the value of $a$.

Question

Let $P : y^2 = 4ax, a > 0$ be a parabola with focus $S$. Let the tangents to the parabola $P$ make an angle of $\frac{\pi}{4}$ with the line $y = 3x + 5$ touch the parabola $P$ at $A$ and $B$. Then the value of $a$ for which $A, B$ and $S$ are collinear is:

Options:

(A) 8 only

(B) 2 only

(C) 1/4 only

(D) any $a > 0$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before the elegant curve of a parabola, defined by the equation $y^2 = 4ax$. This is not just an equation; it is a path, a trajectory, and a fundamental shape of nature.

We are given a reference line, $y = 3x + 5$, which acts as our anchor. By comparing this to the standard form $y = mx + c$, we immediately identify its slope as $m_1 = 3$.

Our task is to find the tangents to the parabola that make an angle of $\frac{\pi}{4}$ with this line.

### The Dance of the Slopes

To find the slopes of these tangents, we invoke the angle formula between two lines:

$$	an(	heta) = \left| \frac{m - m_1}{1 + m \cdot m_1} ight|$$

With $	heta = \frac{\pi}{4}$, we know $	an(	heta) = 1$. Substituting our known slope $m_1 = 3$, we get the equation:

$$1 = \left| \frac{m - 3}{1 + 3m} ight|$$

This absolute value equation splits into two distinct paths. In the first case, $\frac{m - 3}{1 + 3m} = 1$, which simplifies to $m - 3 = 1 + 3m$, leading us to $2m = -4$, or $m = -2$.

In the second case, $\frac{m - 3}{1 + 3m} = -1$, which gives $m - 3 = -1 - 3m$, leading to $4m = 2$, or $m = \frac{1}{2}$. We have found our two slopes: $m_A = -2$ and $m_B = \frac{1}{2}$.

### The Perpendicular Revelation

Now, look closely at these two values. If we multiply them, we get $m_A \cdot m_B = (-2) \cdot \left(\frac{1}{2}ight) = -1$.

This is the moment of truth! The product of the slopes is exactly $-1$, which means our two tangents are **perfectly perpendicular** to each other.

This is not a coincidence; it is a geometric necessity. There is a beautiful, standard property in the study of parabolas: if two tangents to a parabola are perpendicular, their point of intersection must lie on the **directrix** of the parabola.

For our parabola $y^2 = 4ax$, the directrix is the vertical line $x = -a$. Therefore, the point where these two tangents meet sits squarely on this line.

### The Focal Chord Connection

We are almost at the finish line. We have two tangents touching the parabola at points $A$ and $B$. The line segment $AB$ is the chord of contact for the intersection point of the tangents.

Because this intersection point lies on the directrix, we can invoke another powerful property: the chord of contact drawn from any point on the directrix is a **focal chord**.

A focal chord, by definition, must pass through the focus $S(a, 0)$. Since the chord $AB$ is a focal chord, it must contain the focus $S$.

Consequently, the points $A, B,$ and $S$ are **collinear**. This result is independent of the specific value of $a$, provided $a > 0$. The geometry holds true regardless of the parabola's scale.

Select Answer:

Visualized Solution

Visualizing the Parabola Setup

Identifying the Reference Line

Setting up the Angle Formula

Solving for Tangent Slopes (Case 1)

Solving for Tangent Slopes (Case 2)

Analyzing the Product of Slopes

The Directrix Property

The Focal Chord Property

Conclusion and Final Answer

The Sigma Insight: Equation of Tangent and Normal

Analyzing the Setup

The Dance of the Slopes

The Perpendicular Revelation

The Focal Chord Connection

Similar Questions

Comprehension Passage

Length of chord $P Q$ is

If chord $P Q$ subtends an angle $θ$ at the vertex of $y^{2} = 4 a x$ , then $tan θ =$

Comprehension Passage

The value of $r$ is

If $s t = 1$ , then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

The ordinates of the points $P$ and $Q$ on the parabola with focus $(3, 0)$ and directrix $x = - 3$ are in the ratio $3 : 1$ . If $R (α, β)$ is the point of intersection of the tangents to the parabola at $P$ and $Q$ , then $\frac{β ^{2}}{α}$ is equal to

A tangent line $L$ is drawn at the point $(2, - 4)$ on the parabola $y^{2} = 8 x$ . If the line $L$ is also tangent to the circle $x^{2} + y^{2} = a$ , then 'a' is equal to ,

Normals are drawn from the point $P$ with slopes $m_{1}, m_{2}, m_{3}$ to the parabola $y^{2} = 4 x$ . If locus of $P$ with $m_{1} m_{2} = α$ is a part of the parabola itself then find $α$ .

If the common tangent to the parabolas, $y^{2} = 4 x$ and $x^{2} = 4 y$ also touches the circle, $x^{2} + y^{2} = c^{2}$ , then $c$ is equal to:

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?

If two tangents drawn from a point $P$ to the parabola $y^{2} = 4 x$ are at right angles, then the locus of $P$ is

Select Answer:

Visualized Solution

Visualizing the Parabola Setup

Identifying the Reference Line

Setting up the Angle Formula

Solving for Tangent Slopes (Case 1)

Solving for Tangent Slopes (Case 2)

Analyzing the Product of Slopes

The Directrix Property

The Focal Chord Property

Conclusion and Final Answer

The Sigma Insight: Equation of Tangent and Normal

Analyzing the Setup

The Dance of the Slopes

The Perpendicular Revelation

The Focal Chord Connection

Similar Questions

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Length of chord PQ is

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Comprehension Passage

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If chord $P Q$ subtends an angle $θ$ at the vertex of $y^{2} = 4 a x$ , then $tan θ =$

The value of $r$ is

If $s t = 1$ , then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

A tangent line $L$ is drawn at the point $(2, - 4)$ on the parabola $y^{2} = 8 x$ . If the line $L$ is also tangent to the circle $x^{2} + y^{2} = a$ , then 'a' is equal to ,

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