Class 11 Maths Chapter 9 Straight Lines – Exercise 9.2 NCERT Solutions

Introduction

Exercise 9.2 focuses on the angle between two lines and conditions for parallelism and perpendicularity. You will learn how to calculate angles using slopes, check if lines are parallel or perpendicular, and solve problems involving geometric interpretation of line equations.

Key Concepts

1. Slope of Line:

$$m=\frac{y_2-y_1}{x_2-x_1}$$

2. Angle Between Two Lines: If slopes are $m_1,m_2$:

$$\tan \theta =\mid \frac{m_1-m_2}{1+m_1{m}_2}\mid$$

3. Condition for Parallel Lines:

$$m_1=m_2$$

4. Condition for Perpendicular Lines:

$$m_1\cdot m_2=-1$$

NCERT Questions with Solutions (10)

Q1. Find angle between lines $y=2x+3$ and $y=3x+1$.

$$m_1=2,m_2=3$$

$$\tan \theta =\mid \frac{2-3}{1+6}\mid =\frac{1}{7}$$

$\theta ={\tan }^{-1}\left(\frac{1}{7}\right)$

Q2. Find angle between lines $y=x$ and $y=-x$.

$$m_1=1,m_2=-1$$

$$\tan \theta =\mid \frac{1-(-1)}{1+(-1)}\mid =\mid \frac{2}{0}\mid \to \theta ={90}^{\circ }$$

Q3. Check if lines $y=2x+5$ and $y=2x-3$ are parallel. Both have slope $m=2$. Hence parallel.

Q4. Check if lines $y=-\frac{1}{2}x+4$ and $y=2x-1$ are perpendicular.

$$m_1=-\frac{1}{2},m_2=2$$

$$m_1{m}_2=-1⟹\text{Perpendicular}$$

Q5. Find angle between lines $y=0$ and $x=0$. Slopes: horizontal line $m=0$, vertical line slope undefined. Angle = ${90}^{\circ }$.

Q6. Find angle between lines $y=3x+2$ and $y=-\frac{1}{3}x+1$.

$$m_1=3,m_2=-\frac{1}{3}$$

$$m_1{m}_2=-1⟹\text{Perpendicular}$$

Q7. Find angle between lines $y=4x+7$ and $y=4x-2$. Both slopes $m=4$. Hence parallel.

Q8. Find angle between lines $y=-x+3$ and $y=2x+1$.

$$m_1=-1,m_2=2$$

$$\tan \theta =\mid \frac{-1-2}{1+(-2)}\mid =\mid \frac{-3}{-1}\mid =3$$

$\theta ={\tan }^{-1}(3)$

Q9. Find angle between lines $y=\frac{1}{2}x+4$ and $y=\frac{3}{2}x-1$.

$$m_1=\frac{1}{2},m_2=\frac{3}{2}$$

$$\tan \theta =\mid \frac{\frac{1}{2}-\frac{3}{2}}{1+\frac{3}{4}}\mid =\frac{1}{7}$$

$\theta ={\tan }^{-1}\left(\frac{1}{7}\right)$

Q10. Find angle between lines $y=5x+2$ and $y=-\frac{1}{5}x+3$.

$$m_1=5,m_2=-\frac{1}{5}$$

$$m_1{m}_2=-1⟹\text{Perpendicular}$$

FAQs (10)

1. What is slope of line? Ratio of change in $y$ to change in $x$.

2. What is formula for angle between two lines? $\tan \theta =\mid \frac{m_1-m_2}{1+m_1{m}_2}\mid$.

3. When are two lines parallel? If slopes are equal.

4. When are two lines perpendicular? If product of slopes = $-1$.

5. What is slope of horizontal line? Zero.

6. What is slope of vertical line? Undefined.

7. What is intercept form of line? $\frac{x}{a}+\frac{y}{b}=1$.

8. Can two parallel lines intersect? No.

9. Can two perpendicular lines be parallel? No.

10. Why is this exercise important? It builds skills in analyzing relationships between lines in coordinate geometry.

Conclusion

Exercise 9.2 has 10 solved questions and 10 FAQs that strengthen your understanding of angles between lines, parallelism, and perpendicularity. This builds the foundation for advanced coordinate geometry in Class 11 Maths.

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New Syllabus-Class 11 Maths Chapter 9 Straight Lines – Exercise 9.2 NCERT Solutions