KC Sinha Mathematics Solution Class 11 Chapter 21 Straight Lines Exercise 21.6

 Exercise 21.6

Page no 21.75

Type 1

Question 1

Find the point of intersection of the lines $2 x-3 y+8=0$ and $4 x+5 y=6$.

Question 2

Find the points of intersection of the following pair of lines :

(i) $2 x+3 y-6=0,3 x-2 y-6=0$

(ii) $x=0,2 x-y+3=0$

Question 3

For what value of $m$ the line $m x+2 y+5=0$ will pass through the point of intersection of the lines $x-4 y=3$ and $x+2 y=9$ ?

Question 4

Find the point of intersection of the lines

$y f_{1}=x+a t_{1}^{2} \text { and } y t_{2}=x+a t_{2}^{2}$

Question 5

If the straight line $\frac{x}{a}+\frac{y}{b}=1$ passes through the point of intersection of the lines $x+y=3$ and $2 x-3 y=1$ and is parallel to the line $y=x-6$. Find $a$ and $b$.

Question 6

Find the vertices and the area of the triangle whose sides are $x=y, y=2 x$ and $y=3 x+4$

Question 7

The sides of a triangle are given by $x-2 y+9=0,3 x+y-22=0$ and $x+5 y+2=0$. Find the vertices of the triangle.

Question 8

(i) Find the area of the triangle whose sides are $y+2 x=3,4 y+x=5$ and $5 y+3 x=0 .$

(ii) Find the area of the triangle formed by the lines $y-x=0, x+y=0$ and $x-1=0$.

Question 9

Show that the area of the triangle formed by the three straight lines $y=m_{1} x, y=m_{2} x$ and $y=c$ is equal to $\frac{1}{4} e^{2} \sqrt{11}(\sqrt{3}+1)$, where $m_{1}, m_{2}$ are the toots of the equation

$x^{2}+(\sqrt{3}+2) x+\sqrt{3}-1=0$

Page no 21.73

Question 10

The three sides $A B, B C$ and $C A$ of a triangle are $5 x-3 y+2=0 ; x-3 y-2=0$ and $x+y-6=0$, respectively, Find the equation of the altitude through the vertex $A$.

Question 11

Find the equation of line parallel to the y-axis and drawn through the point of intersection of

(i) $x-7 y+5=0$ and $3 x+y-7=0$

(ii) $x-7 y+5=0$ and $3 x+y=0$

Question 12

Find the coordinates of the foot of perpendicular from a point $(-1,3)$ to the line $3 x-4 y-16=0$

Question 13

Two lines cut on the axis of $x$ intercepts 4 and $-4$ and on the axis of $y$ ' intercepts 2 and 6, respectively. Find the coordinates of their point of intersection.

Question 14

Find the coordinates of the orthocenter of the triangle whose vertices are $(-1,3),(2,-1)$ and $(0,0)$.

Question 15

Find the centroid and incentre of the triangle whose sides have the equations $3 x-4 y=0,12 y+5 x=0$ and $y-15=0$

Question 16

Find the co-ordinates of the incentre of the triangle whose sides are $x=3, y=4$ and $4 x+3 y=12$. Also find the centroid.

Question 17

Find the circumcentre of the triangle whose

(i) sides are $3 x-y+3=0,3 x+4 y+3=0$ and $x+3 y+11=0$

(ii) vertices are $(-2,-3),(-1,0)$ and $(7,-6)$

Question 18

Find the orthocenter of the triangle whose vertices are $(0,0),(6,1)$ and $(2,3)$

Question 19

Two vertices of a triangle are $(4,-3)$ and $(-2,5)$. If the orthocenter of the triangle is $(1,2)$. Prove that the third vertex is $(33,26)$.

Question 20

Find the orthocenter of the triangle the equations of whose sides are $x+y=1$, $2 x+3 y=6,4 x-y+4=0$

Type 2

Question 21

Prove that the following lines are concurrent. Also, find their point of concurrency

$5 x-3 y=1,2 x+3 y=23,42 x+21 y=257$

Question 22

Examine whether the following three lines are concurrent or not. If yes find the point of concurrency

$2 x+3 y-4=0, x-5 y+7=0,6 x-17 y+24=0$

Question 23

Find the value of $m$ so that the straight lines $y=x+1, y=2(x+1)$ and $y=m s+3$ are concurrent.

Question 24

Find the value of $m$ so that the lines $3 x+y+2=0,2 x-y+3=0$ and $x+m y-3=0$ may be concurrent.

Question 25

Find the value of $m$ for which the two lines $m x+(2 m+3) y+m+6=0$ and $(2 m+1) x+(m-1) y+(m-9)=0$ intersect at a point on the $y$-axis.

Question 26

(i) Find the value of $m$ so that lines $y=x+1,2 x+y=16$ and $y=m x-4$ may be concurrent,

(ii) Find the value of \& 50 that the lines $2 x+y-3=0,5 r+h y-3=0$ and $3 x-y-2=0$ are concurrent

Page no 21.75

Question 27

If the three lines

$a x+a^{2} y+1=0, b x+b^{2} y+1=0 \text { and } c x+c^{2} y+1=0$

$\mathrm{se}$ concurrent, show that at least two of the three constants $a, b, c$ are cqual. the the straight lines

Question 28

Show that the straight lines

$(b+c) x+a y+1=0, \quad(c+a) x+b y+1=0 \quad$ and $\quad(a+b) x+c y+1=0$ are concurrent.

Question 29

Given a triangle with vertices $A(-2,3), B(-4,1)$ and $C(2,5)$ find the equations of the medians and show that they meet in one point.

Question 30

The co-ordinates of points $A, B$ and $C$ are $(1,2),(-2,1)$ and $(0,6)$. Verify that the medians of the triangle $A B C$ are concurrent. Also find the co-ordinates of the point of concurrence (centroid).

Question 31

Show that the perpendicular bisectors of the sides of the triangle with vertices (7. 2), $(5,-2)$ and $(-1,0)$ are concurrent. Also find the co-ordinates of the point of concurrence (circumcenter).

Question 32

Prove analytically that the right bisectors of the sides of a triangle are concurred.

Question 33

Prove that the (altitudes) perpendiculars drawn from the vertices to the opposite sides of a triangle are concurrent.