Straight line
EXERCISE 23.1
1. Find the slopes of the lines which make the following angles with the positive direction of x-axis :
(i) \( -\dfrac{\pi}{4} \)
sol: m=tan\( \theta \) =tan(\( -\dfrac{\pi}{4} \)) = -1
(ii) \( \dfrac{2\pi}{3} \)
sol: m=tan\( \theta \) =tan(\( \dfrac{2\pi}{3} \)) = tan(\( \pi-\dfrac{\pi}{3} \)) = \( -\sqrt{3} \)
(iii) \( \dfrac{3\pi}{4} \)
sol: m=tan\( \theta \) =tan(\( \dfrac{3\pi}{4} \)) = tan(\( \pi-\dfrac{\pi}{4} \)) = -1
(iv) \( \dfrac{\pi}{3} \)
sol: m=tan\( \theta \) =tan(\( \dfrac{\pi}{3} \)) = \( \sqrt{3} \)
2. Find the slopes of the line passing through the following points :
(i) (-3,2) and (1,4)
(ii) (\( at^{2}_1~,2at_1 \)) and (\( at^{2}_2~,2at_2 \))
(iii) (3,-5) and (1,2)
3. State whether the two lines in each of the following are parallel, perpendicular, or neither:
(i) Through (5 , 6) and (2 , 3) ; through (9 , -2) and (6 , -5)
(ii) Through (9 , 5) and (-1 , 1) ; Through (3 , -5) and (8 , -3)
(iii) Through (6 , 3) and (1 , 1) ; Through (-2 , 5) and (2 , -5)
(iv) Through (3 , 15) and (16 , 6) ; Through (-5 , 3) and (8 , 2)
4. Find the slope of a line bisecting the first quadrant angle.
5. Using the method of slope, show that the following points are collinear
(i) A (4 , 8) , B (5 , 12) , C(9 , 28)
(ii) A (16 , -18) , B (3 , -6) , C(-10 , 6)
6. What is the value of y so that the line through (3,y) and (2,7) is parallel to the line through (-1,4) and (0,6) ?
7. What can be said regarding a line if its slope is
(i) zero
(ii) positive
(iii) negative
8. Show that the line joining (2 , 3) and (-5 , 1) is parallel to the line joining (7 , -1) and (0 , 3) .
9. Show that the line joining (2 , -5) and (2 , 5) is perpendicular to the line joining (6 , 3) and (1 , 1) .
10. Without using Phythagoras theorem, show that the points A (0 , 4) , B (1 , 2) and C (2 , 3) are the vertices of a right angled triangle.
11. Prove that the points (-4 , -1) , (-2 , -4) , (4 , 0) and (2 , 3) are the vertices of a rectangle.
12. If three points A (h,0) , P (a , b) and B (0 , k) lie on a line , show that: \( \dfrac{a}{h}+\dfrac{b}{k}=1 \)
[HINT : use slope of AP = slope of PB]
13. The slope of a line is double of the slope of another line. If tangents of the angle between them is \( \dfrac{1}{3} \) , find the slopes of the other line.
14. Consider the following population and year graphs:
Find the slope of the line AB and using it, find what will be the population in the year 2010.
15. Without using the distance formula , show that points (- 2 , -1) , ( , 0) , (3 , 3) and (-3 , 2) are the vertices of a parallelogram.
16. Find the angle between the X-axis and the line joining the points (3 , -1) , and (4 , -2) .
EXERCISE 23.2
1. Find the equation of the line making an angle of 150° with the x-axis and the cutting of an intercept 2 from y-axis.
2. Find the equation of the straight line :
(i) with slope 2 and y-intercept 3 .
(ii) with slope \( -\dfrac{1}{3} \) and y-intercept -4 .
3. Find the equations of the bisectors of the angles between the co-ordinate axis.
4. Find the equation of the line which is equidistant from the lines x = -2 and x = 6 .
5. Find the equation of the line equidistant from the lines y = 10 and y = -2 .
6. Find the equation of the line which makes an angle of \( tan^{-1}(3) \) with the x-axis and cut off an intercept of 4 units on negative direction of y-axis .
7. Find the equation of the line that has y-intercept -4 and is parallel to the line joining (2 , -5) and (1 , 2) .
8. Find the equation of the line which is perpendicular to the line joining (4 , 2) and (3 , 5) and cut off an intercept of length 3 on y-axis.
9. Find the equation of the perpendicular to the line segment joining (4 , 3) and (-1 , 1) if it cuts off an intercept -3 from y-axis .
EXERCISE 23.3
1. Find the equation of the straight line passing through the points (6 , 2) and having slope -3 .
2. Find the equation of the straight line passing through the points (-2 , 3) and inclined at an angle of 45° with the x-axis .
3. Find the equation of the straight lines passing through the following pair of points:
i. (0 ,0) and (2 , -2)
ii. (a , b) and (\( a+csin\alpha \) , \( ccos\alpha \))
iii. (0 , -a) and (b , 0)
iv. (a , b) and (a+b , a-b)
v. (\( at_{1} \), \( \dfrac{a}{t_1} \)) and (\( at_2 \) , \( \dfrac{a}{t_2} \))
vi. (\( acos\alpha \) , \( asin\alpha \)) and (\( acos\beta \) , \( asin\beta \))
4. Find the equations to the sides of the triangles the coordinates of whose angular points are respectively
(i) (1 , 4) , ( 2, -3) and (-1 , -2)
(ii) (0 , 1) , (2 , 0) and (-1 , -2)
5. Find the equations of the medians of a triangle, the coordinates of whose vertices are
(-1 , 6) , (-3 , -9) and (-1 , -2) .
6. Find the equations of the diagonals of the rectangle the equations of whose sides are x=a , x=a’ , y=b and y=b’ .
7. Find the equation of the bisector of angle A of the triangle whose vertices are A (4 , 3) , B (0 ,0) and C (2 , 3) .
8. The mid-points of the sides of a triangle are (2 , 1) , (-5 , 7) and (-5 , -5). Find the equations of the sides.
9. Find the equations to the straight line which bisects the distance between the points (a , b) , (a’ , b’) and also bisects the distance between the points (-a , b) and (a’ , b’) .
10. Find the equation of the straight line which divides the join of the points (2 , 3) and (-5 , 8) in the ratio 3 :4 and is also perpendicular to it.
11. Prove that the line y-x+2=0 divides the join of points (3 , -1) and (8 , 9) in the ratio 2 : 3 .
12. Find the equations to the straight lines which go through the origin and bisect the portion of the straight line 3x+y=12 which is intercepted between the axes of coordinates.
13. Prove that the perpendicular drawn from the point (4 , 1) on the join of (2 , -1) and (6 , 5) divides it in the ratio 5 : 13 .
14. In what ratio is the line joining the points (2 , 3) and (4 , -5) divided by the line passing through the points (6 , 8) and (-3 , -2) .
15. Find the equations to the altitudes of the triangle whose angular points are A (2 , -2) , B (1 , 1) and C (-1 , 0) .
16. The vertices of a quadrilateral are A (-2 , 6) , B (1 , 2) , C (10 , 4) and D (7 , 8) . Find the equations of the diagonals .
17. Find the equatioin of the line through the point (0 , 2) making an angle \( \dfrac{2\pi}{3} \) with the positive direction of x-axis . Also , find the equation of line parallel to it and crossing y-axis at a distance of 2 units below the origin.
18. The length L (in centimeters) 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 interns of C.
19. The owner of a milk store finds that he can sell 980 litres milk each week at ₹ 14 per liter and 1220 liters of milk each week at ₹ 16 per liter. Assuming a linear relationship between selling price and demand , how many liters could he sell weekly at ₹17 per liter.
