Question

If the line $3x+4y=p$ makes a triangle of area $24$ sq. units with the co-ordinate axes, find the value of $p$.

Answer 1

Given : equation of line $=$ $3x+4y=p$

By intercept theorem, we know,

$\frac{x}{a}+\frac{y}{b}=1$ ....(i)

Therefore, $\frac{3x}{p}+\frac{4y}{p}=1$

$\Rightarrow \frac{x}{p/3}+\frac{y}{p/4}=1$ .....(ii)

On comparing (i) and (ii), we get,

$x$-intercept $\left(OA\right)=$ $\frac{p}{3}$ units

$y$-intercept $\left(OB\right)=$ $\frac{p}{4}$ units

Given: Area of the triangle $=$ $24$ sq. units

$\therefore \frac{1}{2}\times OA\times OB=24$

$\Rightarrow \frac{1}{2}\times \frac{p}{3}\times \frac{p}{4}=24$

$\Rightarrow p^2=24\times 24$

$\Rightarrow p=24$