Problem 54 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The equation of the line passing through the point ( 2 , 3) and making intercept of length 2 units between the lines \(y+2 x=3\) and \(y+2 x=5\), is (A) \(x=2\) (B) \(3 x+4 y=18\) (C) \(4 x+3 y=18\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (D) none of these.

Step 1: Understanding the Given Lines

First, let's understand the lines given: \( y + 2x = 3 \) and \( y + 2x = 5 \). These lines have the same slope, making them parallel.

Step 2: Distance Between Parallel Lines

The distance \( d \) between two parallel lines \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) can be calculated using the formula: \[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]. Substituting the values, we have \( A = 2, B = 1, C_1 = -3, C_2 = -5 \). Thus, the distance is \( \frac{|5 - 3|}{\sqrt{2^2 + 1^2}} = \frac{2}{\sqrt{5}} \).

Step 3: Check the Required Intercept

We need a line passing through (2, 3) that makes an intercept of 2 units between the given parallel lines. The intercept given by the problem is to be determined by the perpendicular distance formula.

Step 4: Equation of the Line with Given Conditions

The perpendicular distance between the point \((x_1, y_1)\) and the line \( Ax + By + C = 0 \) is given by \( \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \). We must find \( C \) so that the perpendicular distance equals 2. For the line through \((2, 3)\) with form \( Ax + By + C \):\( 2*2 + 1*3 + C = 0 \Rightarrow |4 + 3 + C| = 0 \Rightarrow C = -7 \).

Step 5: Verifying the Line with Given Options

Now, check which option corresponds to the line with \( A = 2, B = 1, C = -7 \). The line equation becomes \( 2x + y = 7 \).None of the given options matches this equation directly, implying it could be rearranged or simplified, so let's check each. However, none of the answers correspond to this exact line equation when rearranged or simplified to match given forms.

Key Concepts

Equation of Line

In coordinate geometry, an equation of a line gives us a way to describe a line uniquely in a plane. Lines are usually represented in the form \( Ax + By + C = 0 \).
This lets us know about the line's slope and y-intercept easily. The slope \( m \) can be determined from the equation \( y = mx + c \) when rearranged.
For any given point \((x_1, y_1)\) on the line, every point \((x, y)\) satisfying the equation lies on that line as well. Understanding how to interpret the components \(A\), \(B\), and \(C\) is crucial to comprehend how the line behaves or where it lies on a coordinate plane. Knowing the equation of a line helps greatly in finding distances, angles, or intersections with other geometric entities.

Parallel Lines

Lines that share the same slope but have different y-intercepts are known as parallel lines. In other words, parallel lines never meet, no matter how far they are extended.
The equations for parallel lines can be visualized as \( y = mx + c_1 \) and \( y = mx + c_2 \), sharing the same slope \( m \).
They maintain a constant distance from one another. In the exercise, the lines \(y + 2x = 3\) and \(y + 2x = 5\) illustrate parallel lines, as they both have the slope of \(-2\). Being able to determine when lines are parallel is a foundational skill when grappling with coordinate geometry problems.

Distance Between Parallel Lines

To find the distance between two parallel lines, we use a specific formula. This arises because parallel lines are equidistant everywhere. Let the equations of two parallel lines be \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\).
The distance \(d\) between them is given by the formula:

• \(d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}}\)

This formula calculates the perpendicular distance from one line to the other. It's important because it removes ambiguity about where the shortest distance is measured. It helps to know that this distance measurement is always taken perpendicularly between the lines.

Perpendicular Distance Formula

The perpendicular distance formula is used to find the shortest distance from a point to a line. This is because a straight line has infinite points of distance, but the perpendicular one is the shortest.
Given a line with the equation \(Ax + By + C = 0\) and a point \((x_1, y_1)\), the formula for the perpendicular distance \(d\) is:

• \(d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}\)

This formula ensures that the distance is always positive and reflects the closest possible distance to the given line from the point. Mastering this formula helps solve many problems involving points and lines in coordinate geometry.