Question

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

Answer 1

The equation of a line in the intercept form is

$\frac{x}{y}+\frac{y}{b}=1$ $......(i)$
Here, a and b are the intercepts on x and y axes respectively.
It is given that

$a + b = 9 ⇒ b = 9a \,\,\,\, … (ii)$

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

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

$⇒2(\frac{1}{a}+\frac{1}{9-a})=1$

$⇒2(\frac{9-a+a}{a(9-a)}=1$

$⇒\frac{18}{9a-a^2}=1$

$⇒18=9a-a^2$

$⇒a^2-9a+18=0$

$⇒a^2-6a-3a+18=0$

$⇒a(a-6)-3(a-6)=0$

$⇒(a-6)(a-3)=0$

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

$\frac{x}{6}+\frac{y}{3}=1⇒x+2y-6=0$
If a = 3 and b = 9-3 = 6, then the equation of the line is

$\frac{x}{3}+\frac{y}{6}=1⇒2x+y-6=0$