Exercise 21.9
Page no 21.124
Type 1
Question 1
Examine whether the points $(3,-4)$ and $(2,6)$ are on the same or opposite sides of the line $3 x-4 y=8$ ?
Question 2
Prove that the points $(2,-1)$ and $(1,1)$ are on the opposite sides of the straight line $3 x+4 y-6=0$
Question 3
Find the position of the points $(3,4)$ and $(-1,1)$ with respect to the line $6 x+y-1=0$
Question 4
Prove that the points of intersection of the line $x-y=2$ with the parallel lines $2 x+y=7$ and $2 x+y=16$ are on the opposite sides of the line $x+y=5$
Question 5
Which one of the points $(1,1),(-1,2)$ and $(2,3)$ lies on the side of the line $4 x+3 y-5=0$ on which the origin lies ?
Type 2
Question 6
Find the length of the perpendicular from the point $(-3,4)$ to the line $3 x+4 y-5=0$
Question 7
Find the perpendicular distance of the point $(3,-5)$ from the line $4 y=3 x-26$
Question 8
Find the distance of the point $P$ from the line $l$ in the following :
(i) $1: 12 x-5 y-7=0$ and $P:(3,-1)$
(ii) $l: 12(x+6)=5(y-2)$ and $P:(-3,-4)$
(iii) $l: \frac{x}{a}-\frac{y}{b}=1$ and $P:(b, a)$
(iv) $l: 12(x+6)=5(y-2)$ and $P:(-1,1)$
Question 9
Find the distance of the point of intersection of the lines $2 x+3 y=21$ and $3 x-4 y+11=0$ from the line $8 x+6 y+5=0$
Question 10
In the triangle with vertices $A(2,3), B(4,-1)$ and $C(-1,2)$. Find the length of the altitude from the vertex $A$.
Question 11
(i) What are the points on the axis of $x$ whose perpendicular distance from the line $\frac{x}{3}+\frac{y}{4}=1$ is 4 units ?
(ii) What are the points on $y$-axis whose distance from the line $\frac{x}{3}+\frac{y}{4}=1$ is 4 units ?
Question 12
Find the points on $y$-axis whose perpendicular distance from the linc $4 x-3 y-12=0$ is 3
the ling
Question 13
Find the length of the perpendicular drawn from the origin upon the line joining the points $(a, b)$ and $(b, a)$.
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Question 14
Find the length of the perpendicular from the point $(4,-7)$ to the line joining the origin and the point of intersection of the lines $2 x-3 y+14=0$ and $5 x+4 y-7=0$
Question 15
Find the equation of two straight lines which are parallel to $x+7 y+2=0$ and at unit distance from the point $(1,-1)$.
Question 16
Find the equation of the two straight lines parallel to $3 x-4 y=5$ at a unit distance from it.
Question 17
Find the equations of two lines through $(0, a)$ which are at a distance $a$ from the point $(2 a, 2 a)$.
Question 18
Find the equation of the line through the point of intersection of the lines $x-3 y+1=0$ and $2 x+5 y-9=0$ and whose distance from the origin is $\sqrt{5}$.
Question 19
Find the equation of the straight line passing through the point of intersection of the lines $x-y+1=0$ and $2 x-3 y+5=0$ and at a distance $\frac{7}{5}$ from the point $(3,2)$.
Question 20
If the length of the perpendicular from the point $(1,1)$ to the line $a x-b y+c=0$ be 1 , show that $\frac{1}{c}+\frac{1}{a}-\frac{1}{b}=\frac{c}{2 a b}$.
Question 21
Find the length of the perpendicular from the origin to the line joining two points whose coordinates are $(\cos \theta \cdot \sin \theta)$ and $(\cos \phi \cdot \sin \phi)$.
Question 22
If $p$ and $p_{1}$ be the lengths of the perpendiculars drawn from the origin upon the straight lines $x \sin \theta+y \cos \theta=\frac{1}{2} a \sin 2 \theta$ and $x \cos \theta-y \sin \theta=a \cos 2 \theta$, prove that $4 p^{2}+p_{1}^{2}=a^{2}$.
Question 23
3. (i) Prove that the perpendicular distance between the lines $4 x+3 y=11$ and $8 x+6 y=15$ is $\frac{7}{10}$.
(ii) Find the distance between the parallel lines $3 x-4 y+5=0$ and $3 x-4 y+7=0$
Question 24
If the sum of the perpendicular distances of a variable point $P(x, y)$ from the lines $x+y-5=0$ and $3 x-2 y+7=0$ is always 10 , show that $P$ must move on a line.
Question 25
Determine the distance between the following pair of parallel lines
(i) $4 x-3 y-9=0$ and $4 x-3 y-24=0$
(ii) $15 x+8 y-34=0$ and $15 x+8 y+31=0$
(iii) $3 x-4 y+5=0$ and $3 x-4 y+7=0$
(iv) $I(x+y)+p=0$ and $l x+l y-r=0$
Question 26
(i) Prove that the lines $2 x+3 y=19$ and $2 x+3 y+7=0$ are equidistant from the line $2 x+3 y=6$
(ii) Find the equation of the line midway between the parallel lines $9 x+6 y-7=0$ and $3 x+2 y+6=0$
Question 27
Find the distance between the lines $y=m x+c$ and $y=m x+d$.
Question 28
Sides of a square lie on the line $5 x-12 y-65=0$ and $5 x-12 y+26=0$. Find the area of the square.
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Question 29
The equation of two sides of a square whose area is 25 square thites inte $3 x-4 y=0$ and $4 x+3 y=0$ Find the equation of the other two sides of tre square. Give all possible answers.
Prove that the diagonals of
Question 30
Prove that the diagonals of the parallelogram formed by the lines $\sqrt{3} x+y=0, \sqrt{3} y+x=0, \sqrt{3} x+y=1$ and $\sqrt{3} y+x=1$ are at right angles.
Question 31
Show that the parallelogram formed by $a x+b y+c=0, a p_{1} x+b$ angics. $a x+b y+c_{1}=0$ and $a_{1} x+b_{1} y+c_{1}=0$ will be a rhombus if $a^{2}+b^{2}=a_{1}^{2}+b^{2}$
Question 32
Prove that the diagonals of the parallelogram formed by the four lines $\frac{x}{a}+\frac{y}{b}=1, \frac{x}{b}+\frac{y}{a}=1, \frac{x}{a}+\frac{y}{b}=-1$ and $\frac{x}{b}+\frac{y}{a}=-1$ are perpendicular to each other.
Question 33
The equation of one side of a rectangle is $3 x-4 y-10=0$ and the co-ordinates: of two of its vertices are $(-2,1)$ and $(2,4)$. Find the area of the rectangle and the equation of that diagonal of the rectangle which passes through the point $(2,4)$.
Question 34
Prove that the lines $a x \pm b y \pm c=0$ enclose a rhombus whose area is $\frac{2 c^{2}}{|a b|}$
Question 35
Prove that the product of the lengths of the perpendiculars from the point $\left(\sqrt{a^{2}-b^{2}}, 0\right)$ and $\left(-\sqrt{a^{2}-b^{2}}, 0\right)$ to the line $\frac{x}{a} \cos \theta+\frac{y}{b} \sin \theta=1$ is $b^{2}$.
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