Question

Assertion :The line $3x+2y=24$ meets the $y$-axis at $A$ and $x$-axis at $B$, and the perpendicular bisector of $AB$ meets the line through $(0,-1)$ parallel to the $x$-axis at $C$. The area of the triangle $ABC$ is $91$ square units Reason: Area of the triangle with vertices $(a,0),(0,b)$ and $(a,b)$ is $\frac{ab}{2}$ units

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Reason is True.
For assertion, $A(0,12),B(8,0)$. $D$, the mid-point of $AB$ is $(4,6)$
Equation of the perpendicular bisector is $2x-3y=2\times 4-3\times 6=-10$
Which meets the line $y=-1$ at $C\equiv (-\frac{13}{2},-1)$.

Area of the $△ABC=\frac{1}{2}AB\times CD$
$=\frac{1}{2}\sqrt{64+144}\times \sqrt{{(4+\frac{13}{2})}^2+{(6+1)}^2}$
$=\frac{1}{2}\times 4\sqrt{13}\times 7\sqrt{{\left(\frac{3}{2}\right)}^2+1}=13\times 7=91$
The reason is true but does not explain assertion.