KC Sinha Mathematics Solution Class 11 Chapter 20 Cartesian System of Rectangular Coordianates Exercise 20.4

 Exercise 20.4

Page no 20.37

Question 1

Find the area of the triangle whose vertices are :

(i) $(3,-4),(7,5),(-1,10)$

(ii) $\left(a t_{1}^{2}, 2 a t_{1}\right),\left(a t_{2}^{2}, 2 a t_{2}\right),\left(a t_{3}^{2}, 2 a t_{3}\right)$

(iii) $(a \cos \alpha, b \sin \alpha),(a \cos \beta, b \sin \beta),(a \cos \gamma, b \sin \gamma)$

Question 2

Find the area of the quadrilateral whose vertices are :

(i) $(1,1),(7,-3),(12,2)$ and $(7,21)$;

(ii) $(-4,5),(0,7),(5,-5)$ and $(-4,-2)$

Question 3

Find the area of the pentagon whose vertices are $(4,3),(-5,6),(0,-7)$, $(3,-6),(-7,-2)$.

Question 4

Find the area of the hexagon whose consecutive vertices are $(5,0)$, $(4,2),(1,3),(-2,2),(-3,-1)$ and $(0,-4)$

Question 5

If $A, B, C$ are the points $(-1,5) ;(3,1) ;(5,7)$ respectively and $D, E, F$ are the middle points of $B C, C A$ and $A B$ respectively, prove that

$\triangle A B C=4 \triangle D E F \text {. }$

Question 6

Three vertices of a triangle are $A(1,2), B(-3,6)$ and $C(5,4)$. If $D, E$ and $F$ are the mid-points of the sides opposite to the vertices $A, B$ and $C$, respectively, show that the area of triangle $A B C$ is four times the area of triangle $D E F$.

Question 7

Find the area of a triangle $A B C$ if the coordinates of the middle points of the sides of the triangle are $(-1,-2),(6,1)$ and $(3,5)$.

Question 8

The vertices of a $\triangle A B C$ are $A(3,0), B(0,6)$ and $C(6,9)$. A straight line $D E$ divides $A B$ and $A C$ in the ratio $1: 2$ at $D$ and $E$ respectively, prove that

$\frac{\triangle A B C}{\triangle A D E}=9$

Page no 20.38

Question 9

If $(t, t-2),(t+3, t)$ and $(t+2, t+2)$ are the vertices of a triangle, show that it area is independent of  t.

Question 10

If $A(x, y), B(1,2)$ and $C(2,1)$ are the vertices of a triangle of area 6 square unit show that $x+y=15$ or $-9$

$\left[\mathrm{HO}_{7} \mathrm{~s},\right.$,

Question 11

Prove that the points $(a, b+c),(b, c+a)$ and $(c, a+b)$ are collinear.

Question 12

If the points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ and $\left(x_{3}, y_{3}\right)$ be collinear, show that

$\frac{y_{2}-y_{3}}{x_{2} x_{3}}+\frac{y_{3}-y_{1}}{x_{3} x_{1}}+\frac{y_{1}-y_{2}}{x_{1} x_{2}}=0$

Question 13

If the points $(a, b),\left(a_{1}, b_{1}\right)$ and $\left(a-a_{1}, b-b_{1}\right)$ are collinear, show that $\frac{a}{a_{1}}=\frac{b}{b}$

Question 14

Three points $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$ and $C(x, y)$ are collinear. Prove the $\left(x-x_{1}\right)\left(y_{2}-y_{1}\right)=\left(x_{2}-x_{1}\right)\left(y-y_{1}\right)$.

Question 15

Show that the points $(a, 0),(0, b)$ and $(1,1)$ are collinear if $\frac{1}{a}+\frac{1}{b}=1$.

Question 16

Find the values of $x$ if the points $(2 x, 2 x),(3,2 x+1)$ and $(1,0)$ are collinear.

Question 17

Show that the straight line joining the points $A(0,-1)$ and $B(15,2)$ divides ftc line joining the points $C(-1,2)$ and $D(4,-5)$ internally in the ratio $2: 3$.

Question 18

Find the area of the triangle whose vertices are $(a+1)(a+2),(a+2),(a+2)$ $(a+3),(a+3)$ and $(a+3)(a+4),(a+4)$.

Question 19

The point $A$ divides the join of $P(-5,1)$ and $Q(3,5)$ in the ratio $k: 1$ Find the two values of $k$ for which the area of $\triangle A B C$, where $B$ is $(1,5)$ and $C$ is $(7,-2)$ is equal to 2 units in magnitude.

Question 20

The coordinates of $A, B, C, D$ are $(6,3),(-3,5),(4,-2)$ and $(x, 3 x)$ respectively, If $\frac{\triangle D B C}{\triangle A B C}=\frac{1}{2}$, find $x$.

Question 21

If the area of the quadrilateral whose angular points taken in order are $(1,2)$. $(-5,6),(7,-4)$ and $(h,-2)$ be zero, show that $h=3$.

Question 22

Find the area of the triangle whose vertices $A, B, C$ are $(3,4),(-4,3),(8,6)$ respectively and hence find the length of perpendicular from $A$ to $B C$.

Question 23

The coordinates of the centroid of a triangle and those of two of its vertices are respectively $\left(\frac{2}{3}, 2\right),(2,3),(-1,2)$. Find the area of the triangle.

Question 24

The area of a triangle is 3 square units two of its vertices are $A(3,1), B(1,-3)$ and the centroid of the triangle lies on $x$-axis. Find the coordinates of the third vertex $C$.

Question 25

The area of a parallelogram is 12 square units. Two of its vertices are the points $A(-1,3)$ and $B(-2,4)$. Find the other two vertices of the parallelogram, if the point of intersection of diagonals lies on $x$-axis on its positive side.

Question 26

The area of a triangle is $\frac{3}{2}$ square units. Two of its vertices are the point $A(2,-3)$ and $B(3,-2)$, the centroid of the triangle lies on the line $3 x-y-8=0$ Find the third vertex. $C$.

Question 27

Prove that the quadrilateral whose vertices are $A(-2,5), B(4,-1), C(9,1) and$D(3,7)$is a parallelogram and find its area. If$E$divides$A C$in the ratio$2:$prove that$D, E$and the middle point$F$of$B C$ are collinear.