Problem 75 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the three lines \(x-3 y=p, a x+2 y=q\) and \(a x+y=r\) form a right-angled triangle then : [Online April9, 2013] (a) \(a^{2}-9 a+18=0\) (b) \(a^{2}-6 a-12=0\) (c) \(a^{2}-6 a-18=0\) (d) \(a^{2}-9 a+12=0\)

Step-by-Step Solution

Answer

The correct choice is (a) \(a^{2}-9a+18=0\).

Step 1: Identify Slope of Lines

To determine if the lines form a right-angled triangle, we must first find their slopes. For the line equation \(x - 3y = p\), the slope \(m_1\) is 1/3. For the equation \(ax + 2y = q\), solve for \(y\) to get \(y = -\frac{a}{2}x + \frac{q}{2}\) with the slope \(m_2 = -\frac{a}{2}\). Lastly, for the line \(ax + y = r\), the slope \(m_3\) is \(-a\) after rewritten as \(y = -ax + r\).

Step 2: Set Conditions for Perpendicular Lines

For a right-angled triangle, at least one pair of lines must be perpendicular. Lines are perpendicular when the product of their slopes is -1. We evaluate the products of the slopes: \(m_1 \times m_2 = \frac{1}{3} \times -\frac{a}{2} = -\frac{a}{6}\), \(m_2 \times m_3 = -\frac{a}{2} \times -a = \frac{a^2}{2}\), and \(m_3 \times m_1 = -a \times \frac{1}{3} = -\frac{a}{3}\).

Step 3: Determine Conditions for Each Pair

From the products: 1. \(-\frac{a}{6} = -1\) leads to \(a = 6\). 2. \(\frac{a^2}{2} = -1\) gives \(a^2 = -2\) which is not valid. 3. \(-\frac{a}{3} = -1\) results in \(a = 3\). Thus, \(a = 3\) or \(a = 6\).

Step 4: Use Valid Values of \(a\) to Determine Correct Equation

Substitute \(a = 3\) and \(a = 6\) into possible equations:- For the equation \(a^2 - 9a + 18 = 0\): - \(3^2 - 9\times3 + 18 = 0\) verifies true. - \(6^2 - 9\times6 + 18 = 0\) verifies true.Thus, the correct equation representing the conditions is \(a^2 - 9a + 18 = 0\).

Key Concepts

Right-angled Triangle

Understanding the geometry of a right-angled triangle is key when working with line equations in coordinate geometry.
At its essence, a right-angled triangle is one where one of its three interior angles is exactly 90 degrees. This right angle changes the dynamics of the triangle, making it special and easier to work with in many mathematics problems.
To identify a right-angled triangle formed by lines on a graph, you need to confirm that one pair of the intersecting lines is perpendicular (makes a 90-degree angle).

• Check if any two lines are perpendicular by examining the slopes.
• The three lines shouldn't just be randomly intersecting; they must connect to form the three sides of the triangle.

Understanding these principles will help you identify when given lines form the classic right-angled triangle on a graph.

Slopes of Lines

The slope of a line is a measure of its steepness, often denoted as the rate of change along the y-axis against the x-axis.
Finding the slope is straightforward if the line equation is in the slope-intercept form (y = mx + b). Here, "m" represents the slope.
For equations not in this form, rearrange them to solve for "y".

• For a line like \( x - 3y = p \), rearrange to get \( y = \frac{1}{3}x - \frac{p}{3} \), identifying the slope as \( \frac{1}{3} \).
• The slope tells us not only about the angle of the line but also about its direction. A positive slope means the line rises as it moves right, and a negative slope indicates it falls.

Understanding the slope helps you characterize lines, especially when determining if they are perpendicular.

Perpendicular Lines

When two lines are perpendicular, they intersect to form a 90-degree angle. This property is fundamental in coordinate geometry and many practical applications.
The key scientific rule here is that the product of the slopes of two perpendicular lines is -1.
So, if you have two lines with slopes \( m_1 \) and \( m_2 \), they are perpendicular if \( m_1 \times m_2 = -1 \).

• For example, if one line has a slope of \( \frac{1}{3} \), the line that intersects perpendicularly will have a slope of \( -3 \).
• This rule makes it easy to check perpendicularity without graphing the lines.

In exercise problems involving lines creating right angles, verifying the product of their slopes is usually the go-to method.