The locus of a variable point whose distance from (2, 0) is 2/3 times its distance from the line x =

Q: Wait, why is the line $x = -9/2$ called a directrix? I thought directrices were always vertical lines?

In this specific problem, the directrix is indeed a vertical line, which simplifies our math. However, in the general definition of a conic, the directrix can be any line in the plane. The key is that the distance $PM$ is always the perpendicular distance from the point $P$ to that line, regardless of its orientation.

Q: Why does $e < 1$ result in an ellipse? Is there a physical way to visualize this?

Think of $e$ as a 'stretching factor'. When $e = 1$, the point $P$ is equidistant from the focus and the directrix, creating the open, balanced curve of a parabola. When $e < 1$, the distance to the focus is 'shorter' than the distance to the directrix, which forces the curve to 'bend' back towards the focus, eventually closing the shape into an ellipse. It is the geometric manifestation of a 'tighter' constraint.

Q: Do I need to calculate the full equation of the ellipse to answer this?

Absolutely not! The beauty of the eccentricity definition is that it tells you the 'DNA' of the conic section immediately. By simply identifying the value of $e$, you have already determined the shape. Calculating the full equation is a great exercise for practice, but for this question, the classification is determined solely by the value of $e$.

Question

The locus of a variable point whose distance from $(2, 0)$ is $2/3$ times its distance from the line $x = -9/2$ is

Options:

(A) ellipse

(B) parabola

(C) hyperbola

(D) none of these

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a vast, empty plane. You have a fixed anchor point, $S(2, 0)$, and a boundary line, $x = -\frac{9}{2}$.

Now, imagine a point $P(x, y)$ that is bound by a strict, elegant rule: its distance from the anchor $S$ must always be exactly $\frac{2}{3}$ of its distance from the boundary line. This is not just a random movement; this is the birth of a conic section.

### The Core Definition

In the world of coordinate geometry, we define a conic section by this very relationship. We call the fixed point the focus, the fixed line the directrix, and the constant ratio the eccentricity, denoted by $e$.

The equation governing this relationship is:

$$PS = e \cdot PM$$

When you look at the problem statement, you see $PS = \frac{2}{3} \cdot PM$. By simply comparing this to our standard definition, we immediately uncover the identity of our curve: the eccentricity $e$ is $\frac{2}{3}$.

### The Classification of Conics

This is where the magic happens. The value of $e$ acts as a DNA sequence for the curve.

If $e = 1$, the point $P$ is equidistant from the focus and the directrix, tracing the beautiful, open path of a parabola. If $e > 1$, the distance to the focus grows faster than the distance to the directrix, causing the curve to open up into a hyperbola.

But here, we have $e = \frac{2}{3}$. Because $\frac{2}{3} < 1$, the curve is constrained. It cannot escape to infinity; it must loop back on itself. This is the definition of an **ellipse**.

### Why This Matters

You might be tempted to dive into the algebra, squaring both sides and expanding the coordinates to find the full equation of the ellipse. While that is a valid and rewarding path, the JEE Advanced examiner is testing your intuition.

They want to see if you recognize the fundamental property of the conic section. By identifying $e = \frac{2}{3}$, you have already solved the mystery. The path traced by $P$ is, by definition, an **ellipse**.

Take a moment to appreciate the elegance of this. You didn't need to perform complex integration or tedious algebraic expansion to classify the shape. You simply looked at the ratio of distances and understood the geometric soul of the problem.

Keep this perspective in your toolkit. Whenever you see a locus problem involving distances to points and lines, look for the focus, look for the directrix, and find that eccentricity. It is the key that unlocks the most complex problems in coordinate geometry.

Animated Solution for Mathematics - Conic Sections: The locus of a variable point whose distance from $(2, 0)$ is $2/3$ times its distance from the line $x = - 9/2$ is

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Visualized Solution

Plotting the Fixed Point

Plotting the Fixed Line

The Variable Point $P$

Defining the Distances

Mathematical Translation

General Definition of a Conic

Extracting Eccentricity $e$

Conditions for Conics

Identifying the Locus

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

Analyzing the Setup

The Core Definition

The Classification of Conics

Why This Matters

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Animated Solution for Mathematics - Conic Sections: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The locus of a variable point whose distance from (2,0) is 2/3 times its distance from the line x=−9/2 is