Question

The line $3x+2y=24$ meets y-axis at A and x-axis at B. The perpendicular bisector of AB meets the x-axis at C, then area of DABC is

• A. $78$
• B. $97$
• C. $56$
• D. $87$

Answer 1

The correct option is A $78$

Given line

$3x+2y=24$

On putting $x=0$ in given line we get Point $A(0,12)$

On putting $y=0$ in given line we get Point $B(8,0)$

Hence Given line is $AB:3x+2y=24$

Equation of perpendicualr bisector to $AB$ is $2x-3y=\lambda$

Perpendicular bisector bisects line AB in equal parts Hence D is mid-point of AB

$D(4,6)$

D lies on line $2x-3y=\lambda$

Hence $8-18=\lambda \Rightarrow \lambda =-10$

Eqaution of perpendicular bisector be $2x-3y+10=0$

Since perpendicular bisector meets x-axis at C

$2x=-10\Rightarrow x=-5$

$C(-5,0)$

Here if we join B and C triangle ABC is formed with height $h=12$ and base is distance from C to A which is $AC=13$

Area of triangle ABC$=\frac{1}{2}\times (AC\times h)$

$Area=\frac{1}{2}\times 13\times 12$

$Area=78$