KC Sinha Mathematics Solution Class 11 Chapter 21 Straight Lines Exercise 21.7

 Exercise 21.7

Page no 21.91

Type 1

Question 1

Find the angle between the lines $x+3 y-8=0$ and $2 x-3 y+6=0$.

Question 2

Find the obtuse angle between the straight lines $9 x+3 y-4=0$ and $2 x+4 y+5=0$

Question 3

Find the angle between the lines $x=a$ and $b y+c=0$.

Question 4

Find the angle between the lines 3x+5y-2=0

Page no 21.92

Question 5

Find the angle between the lines $\sqrt{3} x+y=1$ and $x+\sqrt{3} y=1$.

Question 6

Find the tangent of the angle between the lines which have intercepts 3,4 and 1 , 8 on the $x$ and $y$-axes respectively.

Question 7

Find the tangent of the angle between the lines whose intercepts on the alt are respectively, $p,-q$ and $q,-p$.

Question 8

Find the angle between the line joining the points $(2,0),(0,3)$ and the line $x+y=1$.

Question 9

The line through $(4,3)$ and $(-6,0)$ intersects the line $5 x+y=0$. Find the angles of intersection.

Question 10

Prove that the lines $\frac{x}{a}+\frac{y}{b}=1$ and $\frac{x}{b}-\frac{y}{a}=1$ are perpendicular to each other .

Question 11

Show that the line joining $(2,-3)$ and $(-1,2)$ is perpendicular to the line joining $(3,7)$ and $(-2,4)$.

Question 12

A line passing through the points $(a, 2 a)$ and $(-2,3)$ is perpendicular to the line $4 x+3 y+5=0$; find the value of $a$.

Question 13

Show that the coordinates of the vertices of an equilateral triangle cannot be al integers.

Question 14

Prove that the line $k^{2} x+k y+1=0$ is perpendicular to the line $x-k y=1$ for a real values $k(\neq 0)$.

Question 15

For what value of $k$ is the line $x-y+2+k(2 x+3 y)=0$ parallel to the line $3 x+y=0 ?$

Question 16

Prove that the lines $2 x-3 y+1=0, x+y=3,2 x-3 y=2$ and $x=4-y$ form a parallelogram.

Question 17

Find the value of $\theta$ between 0 and $\pi$ if $x \cos \theta+y \sin \theta=2$ is perpendicular to the line $x-y=3$.

Question 18

If the line $x-3 y+5+k(x+y-3)=0$, is perpendicular to the line $x+y=1$, find $k$.

Question 19

The line $7 x-9 y-19=0$ is perpendicular to the line through the points $(x, 3)$ and $(4,1)$. Find the value of $x$.

Question 20

Examine which of the following pairs of lines are intersecting, parallel perpendicular and coincident :

(i) $x-2 y+3=0 \quad$ and $\quad 2 x-4 y+5=0$

(ii) $2 x+3 y+5=0$

and $\quad 4 x+6 y+10=0$

(iii) $x-y+1=0$

and $\quad x+y+2=0$

(iv) $x-y+2=0 \quad$ and $\quad 2 x-3 y+5=0$

Type 2

Question 21

(i) Two lines passing through the point $(2,3)$ make an angle of $45^{\circ}$. If the slope of one of the lines is 2 , find the slope of the other.

(ii) Two lines passing through the point $(2,3)$ intersect each other at an angle of $60^{\circ}$. If slope of one line is 2 , find the equation of the other line.

Question 22

Find the slope of the lines which make an angle of $45^{\circ}$ with the line x-2y=3

Question 23

Find the equation of the straight lines passing through (2,-1) and making an angle of 45 with the line  $6 x+5 y=8$

Page no 21.93

Question 24

(i) Find the equation of the legs of a right isosceles triangle if the equation of its hypotenuse is $x-2 y-3=0$ and the vertex of the right angle is at the point $(1,6)$,

(ii) The hypotenuse of a right triangle has its ends at the points $(1,3)$ and $(-4,1)$. Find the equation of the legs (perpendicular sides) of the triangle.

Question 25

Find the equation of the straight lines passing through the origin making an angle $45^{\circ}$ with straight line $\sqrt{3} x+y=11$.

Question 26

Find the equation of the two straight lines through $(1,2)$ forming the two sides of a square of which $4 x+7 y=12$ is one diagonal.

Question 27

A line through the point $P(1,2)$ makes an angle of $60^{\circ}$ with the positive direction of $x$-axis and is rotated about $P$ in the clockwise direction through an angle $15^{\circ}$. Find the equation of the straight line in the new position.

Question 28

Find the equation of the straight lines passing through the origin making an angle $\alpha$ with the straight line $y=m x+c$.

Question 29

A line $x-y+1=0$ cuts the $y$ axis at $A$. This line is rotated about $A$ in the clockwise direction through $75^{\circ}$. Find the equation of the line in the new position.

Question 30

The slope of a line is double of the slope of another line. If tangent of the angle between them is $\frac{1}{3}$, find the slopes of the lines.

Type 3

Question 31

Find the equations of the lines which pass through the point $(4,5)$ and make equal angles with the lines $5 x-12 y+6=0$ and $3 x=4 y+7$.

Question 32

If the lines $y=3 x+1$ and $2 y=x+3$ are equally inclined to the line $y=m x+4$, find the value of $m$.

Question 33

A ray of light passing 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 $A$.

Question 34

let $(2,1),(-3,-2)$ and $(a, b)$ form a triangle. Show that the collection of the points $(a, b)$ form a line for which the triangle is isosceles. Find the equation of that line.