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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