Problem 25 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The straight line \(y=x-2\) rotates about a point where it cuts the \(x\)-axis and becomes perpendicular to the straight line \(a x+b y+c=0 .\) Then, its equation is (A) \(a x+b y+2 a=0\) (B) \(a x-b y-2 a=0\) (C) \(b y+a y-2 b=0\) (D) \(a y-b x+2 b=0\)

Step-by-Step Solution

Answer

The equation is (D) \(a y - b x + 2 b = 0\).

Step 1: Identify the Point of Rotation

The line \(y = x - 2\) intersects the \(x\)-axis when \(y = 0\). Solve for \(x\):\(0 = x - 2\) gives \(x = 2\). Hence, the point is \((2, 0)\).

Step 2: Determine New Slope for Perpendicular Line

Given a line \(ax + by + c = 0\), its slope is \(-\frac{a}{b}\). The perpendicular slope is the negative reciprocal, \(\frac{b}{a}\).

Step 3: Equation of Line Through Rotation Point

For the line \(y = mx + c\) passing through \((x_1, y_1)\), it satisfies \(y_1 = mx_1 + c\). Substitute \((x_1, y_1) = (2,0)\) and \(m = \frac{b}{a}\) into \( y = \frac{b}{a}x + c\). You find \(0 = \frac{b}{a} \cdot 2 + c\) which implies \(c = -\frac{2b}{a}\).

Step 4: Formulate New Line Equation

Substitute \(m = \frac{b}{a}\) and \(c = -\frac{2b}{a}\) back into \(y = \frac{b}{a}x - \frac{2b}{a}\) and rearrange to form \(bx - ay = 2b\).

Step 5: Translate to Standard Linear Form

Rearrange \(bx - ay = 2b\) to the standard form \(-ay + bx = -2b\) which matches option (D) \(a y - b x + 2 b = 0\).

Key Concepts

Equation of a Line

When understanding coordinate geometry, one of the foundational concepts is the equation of a line. Typically, a straight line in a plane can be expressed in the slope-intercept form as \( y = mx + c \). Here, \( m \) represents the slope of the line and \( c \) shows the line's y-intercept, which is where the line crosses the y-axis.
Expressing lines in other forms is possible. The general form of a linear equation is \( ax + by + c = 0 \), where \( a \), \( b \), and \( c \) are constants. This form is quite flexible since it can represent any line in a two-dimensional space.
Finding the equation of a line involves identifying these coefficients. For instance,

• If you know a point on the line and its slope, you can find \( c \) by substituting the coordinates of the point and the slope into the formula.
• If two points on the line are known, you can compute the slope first and then find the y-intercept.

These forms are crucial because they provide different perspectives on the same line. Choosing one form over another depends on the type of problem you're solving.

Perpendicular Lines

In coordinate geometry, perpendicular lines are lines that intersect at a 90-degree angle. A key characteristic of two perpendicular lines is seen in the relationship between their slopes. Simply put, if two lines are perpendicular, the product of their slopes is \(-1\).
This means if one line has a slope \( m \), the line perpendicular to it will have a slope \( -\frac{1}{m} \). This concept is essential when determining or verifying perpendicularity in geometric problems.
Consider this:

• A line with slope \( -\frac{a}{b} \) can have a perpendicular line with slope \( \frac{b}{a} \).
• Knowing the slope of one line can easily help you find the slope of another line perpendicular to it.

Understanding perpendicular lines helps in solving numerous problems involving rectangular shapes or in breaking down spatial relationships in coordinate geometry.

Slope of a Line

The slope of a line in coordinate geometry reflects its steepness and direction. It is usually denoted by \( m \), and calculated as the ratio of the change in y-coordinates to the change in x-coordinates between two points on the line. Mathematically, it’s written as \( m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are points on the line.
The slope provides key information about the line:

• Positive slopes indicate lines that rise as they move from left to right.
• Negative slopes denote lines that fall as they go from left to right.
• If the slope is zero, the line is horizontal and thus parallel to the x-axis.
• Undefined slopes correspond to vertical lines parallel to the y-axis.

The slope not only tells you how steep the line is but also its direction. It's fundamental when drawing lines, solving geometry problems, and computing angles between lines.