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KC Sinha Mathematics Solution Class 11 Chapter 21 Straight Lines Exercise 21.4

 Exercise 21.4

Page no 21.49

Type 1

Question 1

(i) Find the equation of the line whose intercepts on x and y-axes are 2 and -3 respectively.
(ii) Find the equation of the line, which makes intercepts -3 and 2 on the x and y-axes respectively.

Question 2

Find the equation of the straight line which passes through the point (2,3) and cuts off equal intercepts on the axes.

Question 3

Find the equation of the straight line which cuts off equal and positive intercepts from the axes and passes through the point (3,4).

Question 4

Find the equation of the line which cuts off equal and positive intercepts from the axes and passes through the point (\alpha, \beta).

Question 5

Find the equation of the straight line which passes through the point (2,3) and whose intercept on the x-axis is double that on the y-axis.

Question 6

Find the equation of the straight line which passes through the point (2,3) and whose intercept on the y-axis is the thrice that on the x-axis.

Question 7

Find the equation of the straight line passing through the point (3,-4) and cutting off intercepts, equal but of opposite signs, from the axis.

Question 8

A straight line passes through the point (\alpha, \beta) and this point bisects the part of the line intercepted between the axes. Show that the equation of the straight line is \frac{x}{\alpha}+\frac{y}{\beta}=2

Question 9

Find the equation of the straight lines each of which passes through the point (3,2) and cuts off intercepts a and b respectively on the x and y-axes such that a-b=2.

Question 10

(i) Find the equations to the straight lines which pass through the point (-2,3) and cut the axes at A(a, 0) and B(0, b) so that a+b=2.
(ii) Find the equation of the line passing through the point (2,2) and cutting off intercepts on the axes whose sum is 9 .

Question 11

(i) A straight line passes through the point (3,-2) and this point bisects the portion of the line intercepted between the axes; find the equation of the line.
(ii) Point R(h, k) divides a line segment between the axes in the ratio 1: 2. Find the equation of the line.

Question 12

Find the equation of the line which passes through P(1,-7) and meets the axes A and B respectively so that 4 A P-3 B P=0.

Page no 21.50

Question 13

Find the equation to the straight line which passes through the point P(2,6) and cuts the co-ordinate axes at the points A and B respectively, so that \frac{A P}{B P}=\frac{2}{3} )

Question 14

For the straight line \sqrt{3} y-3 x=3 find the intercepts on the x-axis and y-axis

Question 15

Find the equation of the straight line whose intercepts on the axes are twice the intercepts of the straight line 3 x+4 y=6.

Question 16

Find the equation of the straight line passing through (2,1) and bisecting the portion of the straight line 3 x-5 y=15 lying between the axes.

Question 17

Find the equation of the straight lines which pass through the origin and trisect the portion of the straight line 2 x+3 y=6 which is intercepted between the axes.

Question 18

Prove that the points (5,1),(11,4) and (1,-1) lie on a straight line and find its intercepts on the axes and between the axes.

Question 19

Find the intercepts on the axes of the straight line passing through the points (1,-3) and (4,5)

Type 2

Question 20

Find the equation of the line where the perpendicular distance p of the line from origin and the angle \alpha made by the perpendicular with x-axis are given as
(i) p=3 ; \alpha=45^{\circ}
(ii) p=1, \alpha=90^{\circ}
(iii) p=5, \alpha=30^{\circ}
(iv) p=4, \alpha=15^{\circ}

Question 21

The length of the perpendicular from the origin to a line is 7 and the line makes an angle of 150^{\circ} with the positive direction of y-axis. Then find the equation of the line.

Question 22

Find the equation of the straight line upon which the length of the perpendicular from the origin is 2 and this perpendicular makes an angle of 30^{\circ} with the positive direction of y-axis (in clockwise direction).

Question 23

Find the equation of the line which is at a distance 5 from the origin and the perpendicular from the origin to the line makes an angle 60^{\circ} with the positive direction of the x-axis.

Question 24

Find the equation of the straight line upon which the length of the perpendicular from the origin is 6 and the gradient of this perpendicular is \frac{3}{4}.

Question 25

A straight road is at a distance of 5 \sqrt{2} \mathrm{~km} from a place. The shortest distance of the road from the place is in the N.E. direction. Do the following villages which (i) is 6 \mathrm{~km} East and 4 \mathrm{~km} North from the place, lie on the road or not, (ii) is 4 \mathrm{~km} East and 3 \mathrm{~km} North from the place, lie on the road or not?

Type 3

Question 26

(i) Find the coordinates of the point at a distance 6 units from the point (1,1) in the direction making an angle of 60^{\circ} with the positive direction of the x-axis.
(ii) 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 x+y=4 may be at a distance of 3 units from this point.

Page no 21.51

Question 27

(i) Find the distance of the line 2x+y = 3 from the point (-1.3) in the direction whose slope is 1

Question 28

The straight line through P\left(x_{1}, y_{1}\right) inclined at an angle \theta with x-axis meets the line  a x+b y+c=0 in Q. Find the length P Q.

Question 29

A line is drawn through A(4,-1) parallel to the line 3 x-4 y+1=0. Find the coordinates of the two points on this line which are at a distance of 5 units from A.

Question 30

(i) Find the distance of the point (3,5) from the line 2 x+3 y=14 measured parallel to the line x-2 y=1.
(iii) Find the distance of the line 4 x+7 y+5=0 from the point (1,2) along the line 2 x-y=0

Question 31

The co-ordinates of the extremities of one diagonal of a square are (1,1) and (-2,-1). Find the co-ordinates of its other vertices and the equation of the other diagonal.

(IMAGE TO BE ADDED)

Question 32

A B is a side of a regular hexagon A B C D E F and is of length a with A as the origin and A B and A E as the x-axis and y-axis respectively. Find the equation of lines A C. A F and B E.
(IMAGE TO BE ADDED)

Question 33

Find the equations of all sides of the isosceles \triangle A B C and the sides B E and C D of the square B C D E in the figure, where O C=2 units.








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