Exercise 20.2
Page no 20.11
Type 1
Question 1
(i) Find the distance between the following pair of points :
(a) $(0,0),(-5,12)$
(b) $(4,5),(-3,2)$
(ii) Find the distance between points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ when
(a) $P Q$ is parallel to $y$-axis
(b) $P Q$ is parallel to $x$-axis
Page no 20.12
Question 2
Examine whether the points $(1,-1),(-5,7)$ and $(2,5)$ are equidistant from the point $(-2,3)$ ?
Question 3
(i) Find $a$ if the distance between $(a, 2)$ and $(3,4)$ is 8 .
(ii) A line is of length 10 units and one of its ends is $(-2,3)$. If the ordinate of the other end is 9, prove that the abscissa of the other end is 6 or $-10$
Question 4
Find the distance between the points :
(i) $(a \cos \alpha, a \sin \alpha)$ and $(a \cos \beta, a \sin \beta)$
(ii) $\left(a t_{1}^{2}, 2 a t_{1}\right)$ and $\left(a t_{2}^{2}, 2 a t_{2}\right)$
(iii) $(a-b, b-a),(a+b, a+b)$
(iv) $(\cos \theta, \sin \theta),(\sin \theta, \cos \theta)$
Question 5
Find the point on $x$-axis which is equidistant from the following pair of points :
(i) $(7,6)$ and $(-3,4)$
(ii) $(3,2)$ and $(-5,-2)$
(iii) $(7,6)$ and $(3,4)$
Question 6
Find the point on $y$-axis equidistant from points $(-5,-2)$ and $(3,2)$.
Question 7
Using distance formula, examine whether the following sets of points are collinear?
(i) $(3,5),(1,1),(-2,-5)$
(ii) $(5,1),(1,-1),(11,4)$
(iii) $(0,0),(9,6),(3,2)$
(iv) $(-1,2),(5,0),(2,1)$
Question 8
If $A=(6,1), B \equiv(1,3), C \equiv(x, 8)$, find the value of $x$ such that $A B=B C$
Question 9
Prove that the distance between the points $(a+r \cos \theta, b+r \sin \theta)$ and $(a, b)$ is independent of $\theta$.
Question 10
(i) Use distance formula to show that the points $\left(\operatorname{cosec}^{2} \theta, 0\right),\left(0, \sec ^{2} \theta\right)$ and $(1,1)$ are collinear.
(ii) Using distance formula show that $(3,3)$ is the centre of the circle passing through points $(6,2),(0,4)$ and $(4,6)$. Find the radius of the circle.
Question 11
If the point $\left(x_{4}, y\right)$ is equidistant from the points $(2,3)$ and $(6,-1)$. find the relation between $x$ and $y$.
Question 12
If the point $P(x, y)$ be equidistant from the points $(a+b, b-a)$ and $(a-b, a+b)$, prove that $\frac{a-b}{a+b}=\frac{x-y}{x+y}$.
[HOTS
Question 13
Prove that the points $(3,4),(8,-6)$ and $(13,9)$ are the vertices of a right angled triangle.
Question 14
Determine the type (isosceles, right angled, right angled isosceles, equilateral, scalene) of the following triangles whose vertices are :
(i) $(1,1),(-\sqrt{3}, \sqrt{3}),(-1,-1)$
(ii) $(0,2),(7,0),(2,5)$
(iii) $(-2,5),(7,10),(3,-4)$
(iv) $(4,4),(3,5),(-1,-1)$
(v) $(1,2 \sqrt{3}),(3,0),(-1,0)$
(vi) $(0,6),(-5,3),(3,1)$
Question 15
If $A\left(a t^{2}, 2 a t\right), B\left(\frac{a}{t^{2}},-\frac{2 a}{t}\right)$ and $C(a, 0)$ be any three points, show that $\frac{1}{A C}+\frac{1}{B C}$ is independent of $t$.
Question 16
If two vertices of an equilateral triangle be $(0,0)$ and $(3, \sqrt{3})$. find the co-ordinates of the third vertex.
Question 17
Find the circum-center and circum-radius of the triangle Who Vertices are $(-2,3),(2,-1)$ and $(4,0)$.
Page no 20.13
Question 18
If the line segment joining the points $A(a, b)$ and $B(c, d)$ subtends a right angle at the origin, show that $a c+b d=0$.
Question 19
The vertices of a triangle $A B C$ are $A(0,0), B(2,-1)$ and $C(9,2)$, find $\cos B$.
Question 20
If the line segment joining the points $A(a, b)$ and $B(a,-b)$ subtends an angle $\theta$ at the origin, show that $\cos \theta=\frac{a^{2}-b^{2}}{a^{2}+b^{2}}$.
Question 21
The center of a circle is $(2 x-1,3 x+1)$ and radius is 10 units. Find the value of $x$ if the circle passes through the point $(-3,-1)$.
Question 22
Prove that the points $(4,3),(6,4),(5,6)$ and $(3,5)$ are the vertices of a square.
Question 23
Prove that the points $(3,2),(6,3),(7,6)$ and $(4,5)$ are the vertices of a parallelogram. is it a rectangle ?
Question 24
Prove that the points $(6,8),(3,7),(-2,-2),(1,-1)$ are the vertices of a parallelogram.
Question 25
Prove that the points $(4,8),(0,2),(3,0)$ and $(7,6)$ are the vertices of a rectangle.
Question 26
Show that the points $A(1,0), B(5,3), C(2,7)$ and $D(-2,4)$ are the vertices of a rhombus.
Question 27
$A(-4,0)$ and $B(-1,4)$ are two given points. $C$ and $D$ are points which are symmetric to the given points $A$ and $B$ respectively with respect to y-axis. Calculate the perimeter of the trapezium $A B D C$.
Question 28
A line segment $A B$ through the point $A(2,0)$ which makes an angle of $30^{\circ}$ with the positive direction of x-axis is rotated about $A$ in anticlockwise direction through an angle of $15^{\circ}$. If $C$ be the new position of point $B(2+\sqrt{3}, 1)$, find the coordinates of $C$.
Question 29
The point $\left(l_{4}-2\right)$ is reflected in the $x$-axis and then translated parallel to the positive direction of $x$-axis through a distance of 3 units, find the coordinates of the point in the new position.
The line segment joining $A(3,0)$ and $B(5,2)$ is rotated about $A$ in the anticlockwise direction through an angle of $45^{\circ}$ so that $B$ goes to $C$. If $D$ is the reflection of $C$ in y-axis, find the coordinates of $D$.
Type 2
Question 31
If $A B C D$ be a rectangle and $P$ be any point in the plane of the rectangle, then prove that
$P A^{2}+P C^{2}=P B^{2}+P D^{2}$
[Hint : Take $A$ as the origin and $A B$ and $A D$ as $x$ and $y$ axis respectively. Let $A B=a, A D=b]$
$12 \mathrm{~B}$
Question 32
Prove analytically that the diagonals of a rectangle are equal.
Question 33
Prove analytically that the sum of squares of the diagonals of a rectangle is equal to the sum of squares of its sides
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