Problem 147 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the co-ordinates of two points \(\mathrm{A}\) and \(\mathrm{B}\) are \((\sqrt{7}, 0)\) and \((-\sqrt{7}, 0)\) respectively and \(\mathrm{P}\) is any point on the conic, \(9 x^{2}+16 y^{2}=144\), then \(\mathrm{PA}+\mathrm{PB}\) is equal to: (a) 16 (b) 8 (c) 6 (d) 9

Step-by-Step Solution

Answer

The sum \( PA + PB \) is 8.

Step 1: Understand the Problem

We are given two points \( A(\sqrt{7}, 0) \) and \( B(-\sqrt{7}, 0) \) and a conic equation \( 9x^2 + 16y^2 = 144 \). We need to find the sum of the distances from any point \( P \) on this conic to \( A \) and \( B \).

Step 2: Identify the Type of Conic

The given conic equation can be rewritten as \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \), which is the equation of an ellipse centered at the origin with semi-major axis 4 along the x-axis and semi-minor axis 3 along the y-axis.

Step 3: Determine the Foci of the Ellipse

For an ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) with \( a^2 > b^2 \), the foci are located at \( (\pm c, 0) \) where \( c = \sqrt{a^2 - b^2} \). Here, \( a = 4 \) and \( b = 3 \), so \( c = \sqrt{16 - 9} = \sqrt{7} \). Therefore, the foci are \((\sqrt{7}, 0)\) and \((-\sqrt{7}, 0)\), which are the points \( A \) and \( B \).

Step 4: Apply Property of Ellipse

A known property of an ellipse is that for any point \( P \) on the ellipse, the sum of the distances from \( P \) to the two foci is constant and equals the length of the major axis. Here, the major axis length is \( 2a = 8 \).

Step 5: Conclude the Result

Since \( A \) and \( B \) are the foci of the ellipse, and \( P \) lies on the ellipse, the value of \( PA + PB \) is \( 8 \).

Key Concepts

Conic Sections

Conic sections are fundamental shapes that emerge when a plane intersects a cone in various ways. These shapes include circles, ellipses, parabolas, and hyperbolas. Each has unique properties and equations that define it. The shape we are focusing on here is the ellipse, a curved shape distinct for its oval appearance.

Ellipses can be thought of as stretched circles. They have two fixed points known as foci. The defining property of an ellipse is that the sum of the distances from any point on the ellipse to these two foci is constant. This characteristic is distinct to ellipses among conic sections.

In our exercise, the conic is an ellipse because the equation fits the form of an ellipse equation. Identifying this is crucial before solving the problem. The equation of an ellipse is typically written as \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) is the semi-major axis, and \( b \) is the semi-minor axis. This equation dictates the shape and size of the ellipse.

Distance Formula

The distance formula is essential for calculating the distance between two points in a coordinate plane. Given two points, \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(d\) between them is \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].

This formula is crucial when dealing with conic sections as it allows us to determine various distances needed to study the behavior of these shapes. For example, in our problem, we use the distance formula to determine the lengths \( PA \) and \( PB \).

Understanding this formula is vital for solving problems involving foci and any general two-dimensional point measurement in geometry. It ensures precise calculations for determining the relationships between points, lines, and curves on a plane.

Foci of Ellipse

The foci of an ellipse are two fixed points that are vital to its geometric definition. These points determine the core property of an ellipse: for any point on the ellipse, the total distance to both foci remains constant. This distance is equivalent to the length of the major axis.

To find the foci of an ellipse, we use the equation \[ c = \sqrt{a^2 - b^2} \], where \( c \) represents the distance from the center to each focus, \( a \) is the semi-major axis, and \( b \) is the semi-minor axis. For our exercise, we identified the foci at \( (\sqrt{7}, 0) \) and \( (-\sqrt{7}, 0) \) after rearranging the ellipse’s equation.

Understanding the location and significance of the foci is key to understanding and solving problems involving ellipses, as they are directly related to the ellipse's most distinctive property. This concept often proves critical when solving geometry problems that involve the sum of distances or the alignment of points relative to the conic section.