Problem 49 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Consider a curve \(a x^{2}+2 h x y+b y^{2}=1\) and a point \(P\) not on the curve. A line drawn from the point \(P\) intersects the curve at points \(Q\) and \(R\). If the product \(P Q . P R\) is independent of the slope of the line, then the curve is (A) an ellipse (B) a hyperbola (C) a circle (D) none of these

Step-by-Step Solution

Answer

The curve is a circle (C).

Step 1: Understand the Equation

The equation given is \( a x^{2}+2 h x y+b y^{2}=1 \), which is a general equation of a conic section. Our task is to identify when this conic section is a particular type, given a condition regarding points and lines drawn from a point not on the curve.

Step 2: Interpret the Given Condition

The condition states that the product \( P Q \cdot P R \) is independent of the slope of the line. This means that no matter the line's angle, the product of the distances from \( P \) to the points of intersection \( Q \) and \( R \) on the curve remains constant.

Step 3: Relate to Known Conic Properties

For a conic section, if the product of distances from an external point to points of intersection on the curve is constant, this occurs when the conic is a circle. This property is used to define the power of a point related to a circle.

Step 4: Conclude Based on Conic Property

Given the property described earlier for circles, we can conclude that the curve must be a circle if the product \( P Q \cdot P R \) is independent of the slope of the line from \( P \). Because such a property is unique to circles among conic sections, our conclusion aligns with this observation.

Key Concepts

Circle

A circle is a simple closed shape where every point is equidistant from a central point called the center. The fixed distance from the center to any point on the circle is known as the radius. The unique property of a circle is that it maintains constant distance relations when intersected by lines.

In the context of power of a point, if you draw a line from a point outside the circle, intersecting the circle at two points, the product of the distances from the point to each of these intersection points remains the same, regardless of the slope of the line. This intrinsic property of circles makes them easy to identify in such geometric problems.

Ellipse

An ellipse is a stretched-out circle that resembles an oval. It is defined as the set of all points for which the sum of the distances to two fixed points (foci) is constant.

The major axis of an ellipse is the longest diameter, while the minor axis is the shortest. The broader the ellipse, the more its two foci move away from each other, affecting the shape distinctly.

Unlike a circle, the property of a constant product of intersection distances from an external point does not hold for an ellipse. This is because the geometric behavior of an ellipse does not support equal distance relations with an external point through varied line slopes.

Hyperbola

A hyperbola is another type of conic section that consists of two distinct, mirror-image curves. It is characterized by the condition that the difference in distances from any point on the hyperbola to two fixed points (foci) is constant.

Hyperbolas appear frequently in natural phenomena, such as the paths of celestial objects or the shape of certain sound waves.

For hyperbolas, as with ellipses, the product of distances from an external point to intersections with the curve does not remain constant irrespective of line slope. Their unique geometry involves distinct relational properties, unlike those observed in circles.

Power of a Point

The power of a point is a fascinating concept in geometry that relates a point outside a circle to the circle's points of intersection with a line. The power of a point theorem declares that if a point is outside the circle and a line from that point intersects the circle at two points, the product of the segments divided by the line equals the power of the point.

This theorem reinforces the scenario in which the curve in question is a circle, as it is the only conic section where this property—being independent of the slope—is maintained. Understanding this theorem clarifies why circles have such a persistent and consistent nature in analytic geometry.