# Find equation of the line passing through the point (2, 2) and

Question:

Find equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.

Solution:

The equation of a line in the intercept form is

$\frac{x}{a}+\frac{y}{b}=1$ (i)

Here, a and b are the intercepts on x and y axes respectively.

It is given thata $+b=9 \Rightarrow b=9-a \ldots$ (ii)

From equations (i) and (ii), we obtain

$\frac{x}{a}+\frac{y}{9-a}=1$ ...(iii)

It is given that the line passes through point $(2,2)$. Therefore, equation (iii) reduces to

$\frac{2}{a}+\frac{2}{9-a}=1$

$\Rightarrow 2\left(\frac{1}{a}+\frac{1}{9-a}\right)=1$

$\Rightarrow 2\left(\frac{9-a+a}{a(9-a)}\right)=1$

$\Rightarrow \frac{18}{9 a-a^{2}}=1$

$\Rightarrow 18=9 a-a^{2}$

$\Rightarrow a^{2}-9 a+18=0$

$\Rightarrow a^{2}-6 a-3 a+18=0$

$\Rightarrow a(a-6)-3(a-6)=0$

$\Rightarrow(a-6)(a-3)=0$

$\Rightarrow a=6$ or $a=3$

If $a=6$ and $b=9-6=3$, then the equation of the line is

$\frac{x}{6}+\frac{y}{3}=1 \Rightarrow x+2 y-6=0$

If $a=3$ and $b=9-3=6$, then the equation of the line is

$\frac{x}{3}+\frac{y}{6}=1 \Rightarrow 2 x+y-6=0$