KC Sinha Mathematics Solution Class 11 Chapter 20 Cartesian System of Rectangular Coordianates Exercise 20.2

 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.

Question 30

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