Question

Line $AB$ passes through point $(1,2)$ and intersects the positive $x$ and $y$ axes at $A(a,0)$ and $B(0,b)$ respectively. If the area of $△AOB$ is $1$ unit the value of $(2a-b{)}^2$ is

• A. $220$
• B. $240$
• C. $248$
• D. $284$

Answer 1

Answer: (A)

$220$

Equation of $AB$ ( intercept form)

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

$(2,3)$ point is on it]

$\Rightarrow \frac{2}{a}+\frac{3}{b}=1$

$\Rightarrow ab=2b+3a---\left(1\right)$

Area of $△OAB=\frac{1}{2}\times OA\times OB$

$\Rightarrow 11=\frac{1}{2}ab\Rightarrow ab=22\Rightarrow a=\frac{22}{b}$

Squaring eqn $\left(1\right)$, $a^2{b}^2=(2b+3a{)}^2$

$\Rightarrow (ab{)}^2=4b^2+9a^2+12ab$

$\Rightarrow (22{)}^2=4b^2+9a^2+(12\times 22)$

$\Rightarrow 4b^2+9a^2=22(22-12)$

$\Rightarrow 4b^2+9a^2=220\Rightarrow \left(A\right)$