KC Sinha Mathematics Solution Class 11 Chapter 21 Straight Lines Exercise 21.1

 Exercise 21.1 

Page no 21.13

Question 1

What can be said about the inclination of a line having slope :

(ii) not defined

 (ii) positive

Question 2

Find the slope of the line whose inclination is :

(i) 0

(ii) $60^{\circ}$

(iii) $150^{\circ}$

(iv) $45^{\circ}$

Question 3

Find the slope of the line through the points :

(i) $(6,3)$ and $(9,3)$

(ii) $(1,2)$ and $(4,2)$

(iiii) $(0,9)$ and $(-3,0)$

(iv) $(0,-4)$ and $(-6,2)$

(v) $(3,-2)$ and $(3,4)$

(vi) $(3,-2)$ and $(-1,4)$

vii) $(3,-2)$ and $(7,-2)$

Question 4

Show that and $(7,-2)$ $(9,-2)$ and $(6,-5)$ joining $(5,6)$ and $(2,3)$ is parallel to the line through $(9,-2)$ and $(6,-5)$.

Question 5

Show that the line through $(2,-5)$ and $(-2,5)$ is perpendicular to the line through $(6,3)$ and $(1,1)$. Examine

Question 6

Examine whether the two lines in each of the following are parallel Perpendicular or neither parallel nor perpendicular :

(i) through $(-2,6)$ and $(4,8)$; through $(8,12)$ and $(4,24)$

(ii) through $(9,5)$ and $(-1,1)$; through $(8,-3)$ and $(3,-5)$

Page no 21.14

Question 7

$A(5,-3), B(8,2), C(0,0)$ are the vertices of a triangle. Show that the median from $A$ is perpendicular to the side $B C$.

Question 8

(i) Determine $y$ so that the line through $(3, y)$ and $(2,7)$ is parallel to the line through $(-1,4)$ and $(0,6) ?$

(ii) Line through the points $(-2,6)$ and $(4,8)$ is perpendicular to the line through the points $(8,12)$ and $(x, 24)$. Find the value of $x$.

(iii) Find the value of $x$ for which the points $(x,-1),(2,1)$ and $(4,5)$ are collinear.

Question 9

Find the slope of the line, which makes an angle of $30^{\circ}$ with the positive direction of $y$-axis measured anticlockwise.

Question 10

Find the slope of a line, which passes through the origin and the mid-point of the line segment joining the points $A(0,-4)$ and $B(8,0)$.

Question 11

Find the angle between the $x$-axis and the line joining the points $(3,-1)$ and $(4,-2)$

Question 12

A line passes through $\left(x_{1}, y_{1}\right)$ and $(h, k)$. If slope of the line is $m$, show that

$k-y_{1}=m\left(h-x_{1}\right)$

Question 13

Using slopes, show that the points $(1,1),(2,3)$ and $(3,5)$ are collinear.

Question 14

$A(3,4), B(-3,0)$ and $C(7,-4)$ are the vertices of a triangle. Show that the line joining the mid-points of $A B$ and $A C$ is parallel to $B C$ and is half of it.

Question 15

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

Question 16

By using the concept of slope, show that $(-2,-1),(4,0),(3,3)$ and $(-3,2)$ are the vertices of a parallelogram.

Question 17

A quadrilateral has vertices $(4,1)$, $(1,7)$, $(-6,0)$ and $(-1,-9)$. Show that the mid-points of the sides of this quadrilateral form a parallelogram.

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Question 18

Prove that a median of an equilateral triangle is perpendicular to the corresponding side.

[Hint : Let $A B C$ be the equilateral triangle. Take $B C=2 k$ and middle point of $B C$ as the origin, $B C$ as $x$-axis and the line perpendicular to $B C$ as $y$-axis. Let $A \equiv(\alpha, \beta)]$

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Question 19

Prove that the diagonals of a rhombus are at right angles.

[Hint : Let the rhombus be $O A B C$. Take $O$ as the origin and $O A$ as $x$-axis.

Let $A \equiv(h, 0), C \equiv(\alpha, \beta) .$ Then $B \equiv(\alpha+h, \beta)]$

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