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KC Sinha Mathematics Solution Class 11 Chapter 21 Straight Lines Exercise 21.8

 Exercise 21.8

Page no 21.105

Type 1

Question 1

Find the equation of the straight line parallel to x+2y= 3 and passing through the point (3,4)

Question 2

Find the equation of the through (-2, 3) parallel to the line 3x-4y +2=0

Question 3

Find the equation of the line through (-2,1) and parallel to line x = 0

Question 4

Find the equation to the straight line parallel to 3x-4y+6= 0 and passing through the middle point of the join of points (2,3) and (4,-1)

Question 5

Find the equation to the straight line passing through the point (2,1) and parallel to the line joining the points (2,3) and (3,-1).

Question 6

Find the equation of the straight line which passes through the point (\alpha, \beta) and is parallel to the line l x+m y+n=0

Question 7

Find the equation of the line that has y-intercept 4 and is parallel to the line  2 x-3 y=7

Page no 21.106

Question 8

Prove that the line through the point \left(x_{1}, y_{1}\right) and parallel to the line A x+B y+C=0 is A\left(x-x_{1}\right)+B\left(y-y_{1}\right)=0.

Question 9

Find the equation of a straight line parallel to 2 x+3 y+11=0 and which is such that the sum of its intercepts on the axes is 15 .

Question 10

Find the equation of the line through point (-2,-1) and perpendicular to the line y=x.

Question 11

(i) Find the equation of the straight line passing through the point (2,5) and perpendicular to the line 2 x+5 y=31.
(ii) Find the equation of a line perpendicular to the line x-2 y+3=0 and passing through the point (1,-2)
(iii) Find the equation of the line perpendicular to the line x-7 y+5=0 and having x-intercept 3 .
(iv) Find the equation of a line drawn perpendicular to the line \frac{x}{4}+\frac{y}{6}=1 through the point, where it meets y-axis.

Question 12

Find the equation of the straight line perpendicular to the line 7 x+2 y+7=0 and passing through the origin.

Question 13

Find the equation of the straight line through the point (\alpha, \beta) and perpendicular to the line l x+m y+n=0

Question 14

Find the equation of the straight line through (a \cos \theta, b \sin \theta) perpendicular to the line \frac{x}{a} \cos \theta+\frac{y}{b} \sin \theta=1.

Question 15

Find the equation to the line through the point (-4,-3) and perpendicular to the line joining the points (1,3) and (2,7).

Question 16

Find the equation of the perpendicular bisector of the line segment joining the origin and the point (4,6).

Question 17

Find the equation of the line which divides the line joining the points (2,3) and (-5,8) internally in the ratio of 3: 4 and is perpendicular to it.

Question 18

Find the equation of the straight line perpendicular to 2 x-3 y=5 and cutting off an intercept 1 on the x-axis.

Question 19

Find the equation of the straight line through \left(x_{1}, y_{1}\right) perpendicular to the line joining \left(x_{2}, y_{2}\right) and \left(x_{3}, y_{3}\right)

Question 20

Find the equation of the line that has y-intercept -3 and is perpendicular to the line 3 x+5 y=4

Question 21

Find the equation of a straight line drawn perpendicular to the line \frac{x}{a}+\frac{y}{b}=1 through the point where it meets the y-axis.

Question 22

(i) Find the coordinates of the foot of the perpendicular drawn from the point P(1,-2) on the line 2 x-y+1=0
(ii) Find the coordinates of the foot of perpendicular from the point (-1,3) to the line 3 x-4 y-16=0

Question 23

Find the projection of the point (1,0) on the line joining the points P(-1,2) and Q(5,4)

Page no 21.107

Question 24

(i) Find the image of the point (1,-2) with respect to the line mirror 2 x-y+1=0 
(ii) Assuming that straight lines work as the plane mirror for a point, find the image of the point (1,2) in the line x-3y+ 4= 0

Question 25

If the image of the point (2,1) with respect to a line mirror be (5,2) find the equation of the mirror .

Question 26

If (k, r) is the foot of the perpendicular from \left(x_{1}, y_{1}\right) to l x+m y+n=0, prove that \frac{x_{1}-h}{l}=\frac{y_{1}-r}{m}=\frac{l x_{1}+m y_{1}+n}{l^{2}+m^{2}}.

Type 2

Question 27

Find the equation of the straight line passing through the point (2,-6) and the point of intersection of the lines 5 x-2 y+14=0 and 2 y=8-7 x.

Question 28

Find the equation of the straight line which passes through the point (1,1) and the point of intersection of the lines 3 x+2 y=0 and x-2 y=0.

Question 29

Find the equation of the line through the point of intersection of x+2 y=5 and x-3 y=7, and passing through the point.
(i) (0,0)
(ii) (0,-1)

Question 30

Find the equation of the line through the intersection of 5 x-3 y=1 and 2 x+3 y-23=0, and perpendicular to the line whose equation is :
(i) x=0
(ii) y=0
(iii) 5 x-3 y-1=0

Question 31

Find the equation of the line through the intersection of lines x+2 y-3=0 and 4 x-y+7=0 and which is parallel to 5 x+4 y-20=0

Question 32

Find the equation of line parallel to the y-axis and drawn through the point of intersection of x-7 y+5=0 and 3 x+y-7=0

Question 33

(i) Find the equation to the straight line which passes through the point of intersection of the straight lines x+2 y=5 and 3 x+7 y=17 and is perpendicular to the straight line 3 x+4 y=10.
(ii) Find the equation to the straight line drawn through the point of intersection of x+2 y+3=0 and 3 x+4 y+7=0 and perpendicular to y-x=8

Question 34

A person standing at the junction (crossing) of two straight paths represented by the equations 2 x-3 y+4=0 and 3 x+4 y-5=0 want to reach the path whose equation is 6 x-7 y+8=0 in the least time. Find the equation of the path that be should follow.
[Hint : Required line is perpendicular to line 6 x-7 y+8=0 and passes through the point of intersection of lines 2 x-3 y+4=0 and 3 x+4 y-5=0 ]

Question 35

Find the equation of the straight line passing through the point of intersection of 2 x+3 y+1=0 and 3 x-5 y-5=0 and equally inclined to the axes.

Question 36

Find the equation of the straight line which passes through the point of intersection of the lines 3 x-y=5 and x+3 y=1 and makes equal and positive intercepts on the axes.

page no 21.108

Question 37

The sides A B and A D of a parallelogram A B C D are 2 x-y+1=0 and x+3 y-10=0 respectively and C is the point (-1,-2). Find the equation of the diagonals A C and B D.

Question 38

Find the equation of the line through the intersection of lines 3 x+4 y=7 and x-y+2=0 and whose slope is 5 .

Question 39

Find the equation of the line through the intersection of the lines 2 x+3 y-4 \geq 1 and x-5 y=7 that has its x-intercept equal to -4.

Question 40

Find the equation of the line passing through the intersection of the lines 4 x+7 y-3=0 and 2 x-3 y+1=0 that has equal intercepts on the axes.

Question 41

Prove that the family of lines represented by
x(1+\lambda)+y(2-\lambda)+5=0,
\lambda being arbitrary, pass through a fixed point. Also find the fixed point.

Question 42

Prove that the line x(a+2 b)+y(a-3 b)=a-b passes through a fixed point for different values of a and b. Also find the fixed point.

Question 43

Prove that the following equations represent a family of lines which pass through a fixed point. Also find the fixed point
(i) (\lambda-1) x+\lambda y=1-3 \lambda
(ii) \lambda x+y=4.

Question 44

Prove that all lines represented by the equation
(2 \cos \theta+3 \sin \theta) x+(3 \cos \theta-5 \sin \theta) y-(5 \cos \theta-2 \sin \theta)=0
pass through a fixed point for all values of \theta. Find the coordinates of that point.

























































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