Chapter 9. STRAIGHT LINES

EXERCISE 9.1

1. Draw a quadrilateral in the Cartesian plane, whose vertices are (– 4, 5), (0, 7), (5, – 5) and (– 4, –2). Also, find its area.

Solution :  Let the quadrilateral ABCD  whose vertices are A(– 4, 5), B(0, 7), C(5, – 5) and D(– 4, –2) respectively .

In figure :

    

Ar(ABCD) = Ar () + Ar ()

In  

Here , ,,

We know that ,

 Area of triangle

In  

Here , , ,

Square units .

2. The base of an equilateral triangle with side  lies along the -axis such that the mid-point of the base is at the origin. Find vertices of the triangle.

Solution : Let PQR be an equilateral triangle .

In figure ,

  

So,  PQ = QR

 

 [Squaring both side]

 

Therefore, the vertices of triangle are () ,( ) and ()  or () ,( ) and () .

3. Find the distance between P and Q when : (i) PQ is parallel to the -axis, (ii) PQ is parallel to the -axis.

Solution : (i) When PQ is parallel to the -axis, then   .

Using distance formula , We have

 

 units

(ii) When PQ is parallel to the -axis, then

Using distance formula , We have

 

 units

4. Find a point on the -axis, which is equidistant from the points (7, 6) and (3, 4).

Solution : Let P() is equidistant from the points A(7, 6) and B(3, 4) .

A/Q,  AP = BP

 

 [Squaring both side]

Therefore, the point on the x-axis is .

5. Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points P (0, – 4) and B (8, 0).

6. Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and (–1, –1) are the vertices of a right angled triangle.
7. Find the slope of the line, which makes an angle of 30° with the positive direction of y-axis measured anticlockwise.
8. Find the value of  for which the points (, – 1), (2,1) and (4, 5) are collinear.
9. Without using distance formula, show that points (– 2, – 1), (4, 0), (3, 3) and (–3, 2) are the vertices of a parallelogram.
10. Find the angle between the -axis and the line joining the points (3,–1) and (4,–2).
11. The slope of a line is double of the slope of another line. If tangent of the angle between them is , find the slopes of the lines.
12. A line passes through ( , ) and (h, k). If slope of the line is m, show that   .
13. If three points  and  lie on a line, show that  .
14. Consider the following population and year graph (Fig 10.10), find the slope of the line AB and using it, find what will be the population in the year 2010 ?

EXERCISE 10.2

In Exercises 1 to 8, find the equation of the line which satisfy the given conditions:
1. Write the equations for the -and -axes.
2. Passing through the point (– 4, 3) with slope .
3. Passing through (0, 0) with slope .
4. Passing through  and inclined with the x-axis at an angle of 75° .
5. Intersecting the -axis at a distance of 3 units to the left of origin with slope –2.
6. Intersecting the -axis at a distance of 2 units above the origin and making an angle of 30° with positive direction of the -axis.
7. Passing through the points (–1, 1) and (2, – 4).

8. Perpendicular distance from the origin is 5 units and the angle made by the perpendicular with the positive -axis is 300.
9. The vertices of ∆ PQR are P (2, 1), Q (–2, 3) and R (4, 5). Find equation of the median through the vertex R.
10. Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6).
11. A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio . Find the equation of the line.
12. Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).
13. Find equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.
14. Find equation of the line through the point (0, 2) making an angle  with the positive -axis. Also, find the equation of line parallel to it and crossing the -axis at a distance of 2 units below the origin.
15. The perpendicular from the origin to a line meets it at the point (–2, 9), find the equation of the line.
16. The length L (in centimetre) of a copper rod is a linear function of its Celsius temperature C. In an experiment, if L = 124.942 when C = 20 and L= 125.134 when C = 110, express L in terms of C.
17. The owner of a milk store finds that, he can sell 980 litres of milk each week at Rs 14/litre and 1220 litres of milk each week at Rs 16/litre. Assuming a linear relationship between selling price and demand, how many litres could he sell weekly at Rs 17/litre?
18.  is the mid-point of a line segment between axes. Show that equation of the line is  .
19. Point  divides a line segment between the axes in the ratio . Find equation of the line.
20. By using the concept of equation of a line, prove that the three points (3, 0), (– 2, – 2) and (8, 2) are collinear.

EXERCISE 10.3

1. Reduce the following equations into slope - intercept form and find their slopes and the  - intercepts.
(i) ,     (ii) ,      (iii) .
2. Reduce the following equations into intercept form and find their intercepts on the axes.
(i) , (ii) , (iii) .
3. Reduce the following equations into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis.
(i) ,     (ii) , (iii)   .
4. Find the distance of the point (–1, 1) from the line
5. Find the points on the -axis, whose distances from the line  are 4 units.
6. Find the distance between parallel lines
(i)  and

(ii)  and .

7. Find equation of the line parallel to the line  and passing through the point (–2, 3).
8. Find equation of the line perpendicular to the line  and having  intercept 3.
9. Find angles between the lines  and   .
10. The line through the points (, 3) and (4, 1) intersects the line  at right angle. Find the value of .
11. Prove that the line through the point ( , ) and parallel to the line  is  .
12. Two lines passing through the point (2, 3) intersects each other at an angle of 60°. If slope of one line is 2, find equation of the other line.
13. Find the equation of the right bisector of the line segment joining the points (3, 4) and (–1, 2).
14. Find the coordinates of the foot of perpendicular from the point (–1, 3) to the line .
15. The perpendicular from the origin to the line  meets it at the point (–1, 2). Find the values of  and .
16. If  and  are the lengths of perpendiculars from the origin to the lines  and , respectively, prove that .
17. In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.
18. If  is the length of perpendicular from the origin to the line whose intercepts on the axes are  and , then show that   .

Miscellaneous Exercise on Chapter 10

1. Find the values of  for which the line  is
(a) Parallel to the -axis,
(b) Parallel to the -axis,
(c) Passing through the origin.
2. Find the values of  and , if the equation  is the normal form of the line .
3. Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are 1 and – 6, respectively.
4. What are the points on the y-axis whose distance from the line  is 4 units.
5. Find perpendicular distance from the origin to the line joining the points  and
6. Find the equation of the line parallel to y-axis and drawn through the point of intersection of the lines  and .
7. Find the equation of a line drawn perpendicular to the line through the point, where it meets the -axis.
8. Find the area of the triangle formed by the lines  and.
9. Find the value of  so that the three lines  and  may intersect at one point.
10. If three lines whose equations are ,   and  are concurrent, then show that .
11. Find the equation of the lines through the point (3, 2) which make an angle of 45° with the line .
12. Find the equation of the line passing through the point of intersection of the lines  and  that has equal intercepts on the axes.

13. Show that the equation of the line passing through the origin and making an angle  with the line  is .
14. In what ratio, the line joining (–1, 1) and (5, 7) is divided by the line   ?
15. Find the distance of the line  from the point (1, 2) along the line .
16. Find the direction in which a straight line must be drawn through the point (–1, 2) so that its point of intersection with the line  may be at a distance of 3 units from this point.
17. The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (– 4, 1). Find an equation of the legs (perpendicular sides) of the triangle.
18. Find the image of the point (3, 8) with respect to the line  assuming the line to be a plane mirror.
19. If the lines  and  are equally inclined to the line , find the value of .
20. If sum of the perpendicular distances of a variable point P () from the lines  and  is always 10. Show that P must move on a line.
21. Find equation of the line which is equidistant from parallel lines   and  .
22. A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.
23. Prove that the product of the lengths of the perpendiculars drawn from the points  and   to the line  is .

24. A person standing at the junction (crossing) of two straight paths represented by the equations  and  wants to reach the path whose equation is  in the least time. Find equation of the path that he should follow.