Problem 85 - A Complete Resource Book in Mathematics for JEE Main

Question

The perpendicular bisector of the line segment joining \(P(1,4)\) and \(Q(k, 3)\) has \(y\)-intercept \(-4\). Then a possible value of \(k\) is (A) 1 (B) 2 (C) \(-2\) (D) \(-4\)

Step-by-Step Solution

Answer

The possible value of \(k\) is \(-4\).

Step 1: Find the midpoint of segment PQ

First, calculate the midpoint of the line segment joining points \(P(1,4)\) and \(Q(k, 3)\). The midpoint \(M(x_m, y_m)\) is given by \(x_m = \frac{1+k}{2}\) and \(y_m = \frac{4+3}{2} = \frac{7}{2}\).

Step 2: Determine the slope of segment PQ

The slope \(m\) of the line segment joining \(P(1,4)\) and \(Q(k, 3)\) is \(m = \frac{3-4}{k-1} = \frac{-1}{k-1}\).

Step 3: Find the slope of the perpendicular bisector

The slope of the perpendicular bisector \(m_\perp\) is the negative reciprocal of the slope of PQ. So, \(m_\perp = k-1\).

Step 4: Use intercept information to write the equation of the bisector

The equation of the perpendicular bisector can be written as \(y - \frac{7}{2} = (k-1)(x - \frac{1+k}{2})\). We know the \(y\)-intercept is \(-4\), so substituting \(x=0\) into the equation and setting \(y = -4\), we have: \(-4 - \frac{7}{2} = (k-1)(0 - \frac{1+k}{2})\).

Step 5: Solve for k

Simplify and solve the equation from the previous step: 1. Move to one side: \(- \frac{15}{2} = -(k-1)\frac{1+k}{2}\)2. Multiply through: \(15 = (k-1)(1+k)\)3. Expand: \(15 = k^2 - 1k + k - 1\)4. Rearrange: \(k^2 - 1 - 15 = 0\)5. Derived quadratic equation becomes \(k^2 - 16 = 0\)6. Solving for \(k\) gives: \(k = \pm \sqrt{16} = \pm 4\). Since the choices given don't have \(+4\), the possible value is \(-4\).

Key Concepts

Perpendicular Bisector

The perpendicular bisector of a line segment is a line that divides the segment into two equal lengths and is perpendicular to the segment itself. This means it forms a right angle with the original segment, standing at 90 degrees to it. It not only bisects, or cuts the segment in half, but also ensures every point on it is equidistant from both endpoints of the original segment. In coordinate geometry, to find the perpendicular bisector of a segment joining two points, it's crucial to calculate the midpoint of the segment, as this point will lie on the bisector. Once the midpoint is identified, the perpendicular bisector will have a slope that is the negative reciprocal of the slope of the segment. This change in slope ensures the bisector is perpendicular. For instance, with the segment joining points \( P(1,4) \) and \( Q(k, 3) \), the perpendicular bisector has a specific slope as it heads through the midpoint, and must satisfy any additional conditions, such as passing through a given \( y \)-intercept, here \(-4\). Understanding how these elements work together allows us to build the equation of the perpendicular bisector.

Midpoint Formula

The midpoint of a line segment in a coordinate plane is a point where the segment is equally divided into two parts. To find this midpoint, you utilize the midpoint formula. This formula takes the average of the \( x \)-coordinates and the \( y \)-coordinates of the points that form the segment.For two points, \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the midpoint \( M(x_m, y_m) \) is determined by:

• \( x_m = \frac{x_1 + x_2}{2} \)
• \( y_m = \frac{y_1 + y_2}{2} \)

This step simplifies to averaging the coordinates. In our problem, with points \( P(1,4) \) and \( Q(k, 3) \), the midpoint is calculated using these averages. For instance, the \( x \)-coordinate of the midpoint is \( x_m = \frac{1+k}{2} \), and the \( y \)-coordinate is \( y_m = \frac{7}{2} \). This midpoint is crucial since the perpendicular bisector will pass through it, helping us frame the equation further.

Slope of a Line

The slope of a line is a measure of its steepness and direction. It's calculated by the change in the \( y \)-coordinates divided by the change in the \( x \)-coordinates of any two distinct points on the line. Mathematically, the slope \( m \) is given by:\( m = \frac{y_2 - y_1}{x_2 - x_1} \)The slope can indicate whether a line is rising or falling as you move from left to right. A positive slope means the line rises, while a negative slope means it falls.For the segment joining \( P(1, 4) \) and \( Q(k, 3) \), the slope is calculated as \( m = \frac{3 - 4}{k - 1} = \frac{-1}{k-1} \). The relationship between slopes of perpendicular lines is crucial here. Specifically, the slope of the perpendicular bisector is the negative reciprocal of this calculated slope of the segment. Hence, if a line's slope is \( m \), the slope of a line perpendicular to it is \( -\frac{1}{m} \). This understanding is essential to formulating the equation of the perpendicular bisector, ensuring the line behaves as expected across the coordinate plane.