Question

Find the equation of lines passing through the point $(2,2)$ such that the sum of their intercepts on the axes is $9$.

Answer 1

Let the equation of the line be $\frac{x}{a}+\frac{y}{b}=1$.....(1). Where $a$ and $b$ are the intercept made by the line from the $x$-axes and $y$-axes respectively.

Then according to the problem, $a+b=9$......(2).

Again (1) passes through $(2,2)$ then we have,

$\frac{2}{a}+\frac{2}{b}=1$.

or, $2(a+b)=ab$

or, $18=a(9-a)$ [Using (2)]

or, $a^2-9a+18$=0

or, $(a-3)(a-6)=0$

So, $a=3,6\Rightarrow b=6,3$.

So two lines are possible and the equation of the lines be

$\frac{x}{3}+\frac{y}{6}=1$ and $\frac{x}{6}+\frac{y}{3}=1$.