Problem 57 - A Complete Resource Book in Mathematics for JEE Main

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 \(P Q \cdot P R\) is independent of the slope of the line, then the curve is a (A) parabola (B) circle (C) ellipse (D) none of these

Step-by-Step Solution

Answer

The curve is a circle.

Step 1: Rewrite the Given Equation

The equation of the curve is given as \( a x^2 + 2hx y + b y^2 = 1 \). This is a general form of a conic section, which can represent a parabola, circle, or ellipse, depending on the values of \(a\), \(b\), and \(h\).

Step 2: Equation of the Line through P

Consider a line with the equation \(y = mx + c\) that intersects the curve. Here, \(m\) is the slope of the line and \(c\) is the y-intercept, related to point \(P\).

Step 3: Substitute Line Equation into Curve Equation

Substitute \(y = mx + c\) into the curve equation: \( a x^2 + 2hx (mx + c) + b (mx + c)^2 = 1 \). This equation represents the x-coordinates of the intersection points \(Q\) and \(R\). Simplifying this, we get a quadratic equation in terms of \(x\).

Step 4: Discriminant Analysis to Determine Curve Type

For the product \(PQ \cdot PR\) (the product of the roots of the quadratic) to be independent of \(m\), the discriminant of the quadratic should be zero or a constant independent of the slope. This property indicates that the conic section could be a circle, as only then is \(PQ \cdot PR\) constant for any line passing through a given point \(P\).

Step 5: Confirm that the Curve is a Circle

The discriminant condition and the independence from the slope suggest that the given conic is a circle. The product of segments being constant across varying slopes and lines is a unique property of circles.

Key Concepts

circle

A circle is a simple closed shape, defined as the set of all points in a plane that are equidistant from a given point, known as the center. The equation for a standard circle centered at the origin \[ x^2 + y^2 = r^2 \]shows that the distance from the center \[ (0,0) \]is equal to the radius, \( r \).
When analyzing conic sections, if we transform coordinates or move the circle's center, the equation becomes more complex, but maintains the squared equality between distance components. For the circle given in the exercise, the equation \( a x^2 + 2 h x y + b y^2 = 1 \)can reduce to that of a circle under certain conditions on coefficients \(a\), \(b\), and \(h\).
Key points to remember about circles:

• Every line that intersects a circle does so in exactly two points or is tangent (intersects at exactly one).
• For a circle, the product of segments from any point outside the circle to the two points of intersection is fixed, indicating balance.

parabola

A parabola is a unique curve defined as the set of all points equidistant from a point called the focus and a line called the directrix. In conic sections, it is usually represented by an equation of the form:\[ y = ax^2 + bx + c \] or sometimes in a rotated form.
Parabolas have some distinct properties that distinguish them from other conic sections:

• They always have a single axis of symmetry, which means every point on the parabola is mirrored along this line.
• The parabola opens either upwards, downwards, or sideways, depending on the orientation and coefficients.
• Lines intersecting a parabola typically do so at one or two points.

When considering the equation from the exercise, if it were to represent a parabola, the conditions would not allow for a constant product of distances from a fixed point to points of intersection on the parabola. Thus, independent slope properties seen here do not apply to parabolas in practice.

ellipse

An ellipse is an elongated circle, described as the set of all points for which the sum of the distances to two fixed points (called foci) is a constant. It can be typically represented in the form:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]for an ellipse centered at the origin with axes aligned with x and y.
Some important features about ellipses include:

• The longer axis of an ellipse is called the major axis, while the shorter is the minor axis.
• Ellipses have two symmetries along each of its axes and are not self-similar like circles.

In the context of conic sections, ellipses share certain properties with circles but differ in their geometry and focal characteristics. The exercise implies a curve where the product of segments through intersection points is constant, leading to a realization that only a circle, and not an ellipse, can maintain that geometric property across variable angles from an outside point.