Let equations of two adjacent sides of a parallelogram A B C D be 2 x-3 y=-23 and 5 x+4 y =23 . If e

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Let the equations of two adjacent sides of a parallelogram $A B C D$ be $2 x-3 y=-23$ and $5 x+4 y$ $=23$. If the equation of its one diagonal $AC$ is $3 x +$ $7 y=23$ and the distance of A from the other diagonal is $d$, then $50 d ^2$ is equal to $........$.

A

$528$

B

$526$

C

$529$

D

$527$

(JEE MAIN-2023)

Solution

$A$ and $C$ point will be $(-4,5)$ and $(3,2)$ mid point of $AC$ will be $\left(-\frac{1}{2}, \frac{7}{2}\right)$ equation of diagonal $BD$ is

$y-\frac{7}{2}=\frac{\frac{7}{2}}{-\frac{1}{2}} \quad\left(x+\frac{1}{2}\right)$

$7 x+y=0$

Distance of $A$ from diagonal $BD$

$= d =\frac{23}{\sqrt{50}}$

$50 d ^2=(23)^2$

$50 d ^2=529$

Standard 11

Mathematics

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