Question

If a, b, c are in G.P. write the area of the triangle formed by the line $ax+by+c=0$ with the coordinates axes.

Answer 1

a,b,c are in G.P.

$\Rightarrow b^2=ac$ ...(1)

Given lines is,

$ax+by+c=0$

$ax+by=-c$

$\frac{x}{-\frac{c}{a}}+\frac{y}{-\frac{c}{b}}=1$

Thus,intercept by line on x axis is -$\frac{c}{a}$

and on y-axis is -$\frac{c}{b}$

Area of $\Delta =\frac{1}{2}(-\frac{c}{a})(-\frac{c}{b})$

$=\frac{1}{2}\left(\frac{c^2}{ab}\right)$

$=\frac{1}{2}\left(\frac{b^2}{\frac{a^2}{ab}}\right)$ [Using (1)]

Area of $\Delta =\frac{1}{2}{\left(\frac{b}{a}\right)}^3$ sq.units