Question

$A$ and $B$ are any two points on the positive $x$ and $y-$axis respectively satisfying $2\left(OA\right)+3\left(OB\right)=10$. If $P$ is the middle point of $AB$ then the locus of $P$ is:

• A. $2x+3y=5$
• B. $2x+3y=10$
• C. $3x+2y=5$
• D. $3x+2y=10$

Answer 1

Answer: (A)

$2x+3y=5$

According to question,
$P$ is the mid point of $AB$.
So, let $P(x,y)$.
Therefore, from mid-point formulai.e.,mid point $(x,y)$ of $(x_1,y_1),(x_2,y_2)$ is

$x-\frac{x_1+x_2}{2},y=\frac{y_1+y_2}{2}$

Hence
$A(2x,0)$ & $B(0,2y)$
And, $OA=2x$ & $OB=2y$
So, locus of $P$ =>$2\left(2x\right)+3\left(2y\right)=10$
=> $2x+3y=5$