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.
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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 $B E$.
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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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