Straight Lines

21

.Here (x₁, y₁) = (–1, 1) and (x₂, y₂) = (2, –4)

Using two point form, equation of line is

$y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

$\Rightarrow y-1=\frac{-4-1}{2-\left(-1\right)}\left(x-\left(-1\right)\right)$

$\Rightarrow y-1=\frac{-5}{2+1}\left(x+1\right)$

$\Rightarrow 3\left(y-1\right)=-5\left(x+1\right)$

$\Rightarrow 3y-3=-5x-5$

$\Rightarrow 5x+3y-3+5=0$

$\Rightarrow 5x+3y+2=0$

20.

The slope of the line is m = tan 30° = $\frac{1}{\mathrm{\surd 3}}$

And x-intercept, c = 2

Using slope intercept form, equation of line is

19.

Given m = –2

x-intercept, d = –3

Using slope intercept form, equation of line is

y = m (x – d)

$\Rightarrow y=\left(-2\right)\left[x-\left(-3\right)\right]$

$\Rightarrow y=\left(-2\right)\left(x+3\right)$

$\Rightarrow y=-2x-6$

$\Rightarrow 2x+y+6=0$

18.

The slope of the line is,  m = tan 75° – tan (45° + 30°)

17.

Given slope = m and $\left(x_0,y_0\right)=\left(0,0\right)$

By point slope from, equation of line is

$m=\frac{y-y_0}{x-x_0}$

$\Rightarrow m=\frac{y-0}{x-0}$

$\Rightarrow m=\frac{y}{x}$

$\Rightarrow mx=y$

$\Rightarrow$ $y=mx$

2.

Given,

$m=\frac{1}{2},\left(x_0,y_0\right)=\left(-4,3\right)$

By point slope form equation of line is

$m=\frac{y-y_0}{x-x_0}$

$\Rightarrow \frac{1}{2}=\frac{\left(y-3\right)}{x-\left(-4\right)}$

$\Rightarrow x+4=2\left(y-3\right)$

$\Rightarrow x+4=2y-6$

$\Rightarrow x-2y+4+6=0$

$\Rightarrow x-2y+10=0$

Exercise 9.2

15. 

For any points on x-axis the y-co-ordinate or the ordinate is always 0. Hence, the x-axis have the equation y = 0.

Similarly for xy points on y-axis the x-co-ordinate or the abscissa is always 0. Hence, the y-axis have the equation x = 0.

14.

From figure,

Slope of AB = $\frac{97-92}{1995-1985}$

$=\frac{5}{10}$

$=\frac{1}{2}$

As, A, B and C lie on same line

Slope of AB = Slope of BC

$\Rightarrow \frac{1}{2}=\frac{a-97}{2010-1995}=\frac{a-97}{15}$

$\Rightarrow 15=2\left(a-97\right)$

$\Rightarrow 15=2a-194$

$\Rightarrow 194+15=2a$

$\Rightarrow 209=2a$

$\Rightarrow a=\frac{209}{2}=104.5$

Hence, population in 2010 will be 104.5 crores.

13.

Since the three points say P (h, o), Q (a, b) and R (o, k) lie on a line.

Slope of PQ = slope of QR

$\Rightarrow \frac{b-o}{a-h}=\frac{k-b}{o-a}$

$\Rightarrow \frac{b}{a-h}=\frac{k-b}{-a}$

$\Rightarrow -ab=\left(k-b\right)\left(a-h\right)$

$\Rightarrow -ab=ka-kh-ab+bh$

$\Rightarrow ka+bh=kh$

Dividing both sides by kh we get,

$\Rightarrow \frac{ka}{kh}+\frac{bh}{kh}=\frac{kh}{kh}$

$\Rightarrow \frac{a}{h}+\frac{b}{k}=1$

12.

Let the line l passes through points (x₁, y₁) and (h, x)

Then slope of l,

$m=\frac{k-y_1}{h-x_1}$

$\Rightarrow \left(h-x_1\right)m=k-y_1$

Hence, proved