Class 11 RD Sharma Solutions- Chapter 23 The Straight Lines- Exercise 23.3

In Class 11, Chapter 23 of RD Sharma's textbook focuses on the concept of straight lines which is a fundamental topic in coordinate geometry. Exercise 23.3 deals with advanced problems involving the properties and equations of straight lines. This exercise helps students understand the practical applications of the line equations and geometric relationships between the different lines.

The Straight Lines

The Straight lines in coordinate geometry are described by their equations which can be used to find various properties such as the slope, intercepts, and distance between the lines. The general form of a line equation is Ax+By+C=0. Through various problems, students explore concepts such as the parallel and perpendicular lines distance from the point to a line and intersections. Mastery of these concepts is crucial for solving complex geometric problems and for applications in higher mathematics.

Question 1. Find the equation of a line making an angle of 150 degrees with the x-axis and cutting off an intercept 2 from the y-axis.

Solution: 

Given, slope, m = tan (150^°) ⇒ m = -1/√3, also, y-intercept = (0,2)

We know that equation, of a line is given as y = mx+ c, m is the slope and c is the intercept that line cuts on y-axis, therefore

equation for the line will be:

y =  -x/√3 + 2

⇒ x - 2√3 + √3y = 0

Question 2. Find the equation of the straight line:

(i) with slope 2 and y-intercept 3;

(ii) with slope -1/3 and y-intercept -4

(iii) with slope -2 and intersecting the x-axis at a distance 3 units to the left of the origin.

Solution: 

(i) We know that equation, of a line is given as y = mx+ c, therefore equation for the line with slope 2 and y-intercept 3 will be: y = 2x + 3

(ii) Similarly, equation of the line with slope -1/3 and y-intercept -4 will be: y = -x/3 - 4

⇒ x +3y +12 = 0

(iii) Since, the line cuts the x-axis at a distance 3 units to the left of origin its coordinate will be (-3,0) and the given slope, m = -2.

Equation of a line passing through a point is given by the formula: y-y₁  = m (x - x₁), hence the equation will be

⇒ y - 0 = -2(x - (-3))

⇒ y = -2x -6

⇒ 2x + y + 6 = 0

Question 3. Find the equations of the bisectors of the angle between the coordinate axes.

Solution:

The equation of the line on the coordinate axes are x=0 and y=0.

The equations of the bisectors of the angle between  x=0 and y=0 are:

x ± y = 0

Question 4. Find the equation of a line which makes an angle of tan^-1(3) with the x-axis and cuts off an intercept of 4 units on negative direction of y-axis.

Solution: 

Here, ∅ = tan^-1 (3) ⇒ m = tan∅ =3, line cuts cuts off an intercept of 4 units on negative direction of y-axis, so the coordinate will be (0, -4)

Hence, the equation of the line is: y = 3x -4

Question 5. Find the equation of a line that has y-intercept -4 and is parallel to the line joining (2, -5) and (1,2).

Solution: 

Since, our required line is parallel to the line passing through the coordinates (2, -5) and (1,2), it will have the same slope as the later line. Therefore, slope, m = (y₂ -  y₁) / (x₂ - x₁) = (2 - (-5) ) / (1 - 2 ) = -7

Also, given y-intercept = -4, hence the required equation of the line is: y = -7x - 4

Question 6. Find the equation of a line which is perpendicular to the line joining (4,2) and (3,5) and cuts off an intercept of length 3 on y-axis.

Solution: 

Slope of the line passing through the points (4,2) and (3,5) is

(y2 -  y1) / (x2 - x1) = (5 - 2 ) / (3 - 4 ) = -3

Now, our required line is perpendicular to the former line, so its slope will be m = 1/3

Also, y-intercept, c = 3, hence the required equation of the line is: y = x/3 + 3

⇒ x - 3y +9 = 0

Question 7. Find the equation of the perpendicular to the line (4,3) and (-1,1) if it cuts off an intercept -3 from y-axis.

Solution: 

Slope of the line passing through the points (4,3) and (-1,1) is

(y2 -  y1) / (x2 - x1) = (1 - 3 ) / (-1 - 4 ) = 2/5

Now, our required line is perpendicular to the former line, so its slope will be m = -5/2

Also, y-intercept, c = -3, hence the required equation of the line is: y = -5x/2 - 3

⇒ 5x + 2y +6 = 0

Question 8. Find the equation of the straight line intersecting y-axis at a distance of 2 units above the origin and making an angle 30° with the positive direction of the x-axis.

Solution: 

Given, m = tan 30° = 1/√3

Since, the line intersects the y-axis at a distance 2 units above the origin its coordinate will be (0,2)

Equation of a line passing through a point is given by the formula: y-y1  = m (x - x₁), hence the equation will be

⇒ y - 2 = 1/√3 . (x - 0)

⇒ √3y - 2√3 = x

⇒ x - √3y + 2√3 = 0

Read More:

• Standard Form of a Straight Line
• How to find the Equation of a Straight Line?

Conclusion

Exercise 23.3 of Chapter 23 provides the rigorous practice for the solving problems related to the straight lines. By working through these exercises students gain a deeper understanding of the line equations and their properties which are essential for both the theoretical and applied mathematics. Consistent practice of these problems enhances problem-solving skills and prepares students for the more advanced topics in the coordinate geometry.