A line intersects the $ y $-axis and $ x $-axis at the points $ \mathrm{P} $ and $ \mathrm{Q} $, respectively. If $ (2,-5) $ is the mid-point of $ P Q $, then the coordinates of $ P $ and $ Q $ are, respectively
(A) $ (0,-5) $ and $ (2,0) $
(B) $ (0,10) $ and $ (-4,0) $
(C) $ (0,4) $ and $ (-10,0) $
(D) $ (0,-10) $ and $ (4,0) $

Given:

A line intersects the \( y \)-axis and \( x \)-axis at the points \( \mathrm{P} \) and \( \mathrm{Q} \), respectively. 

\( (2,-5) \) is the mid-point of \( P Q \).

To do:

We have to find the coordinates of P and Q.

Solution:

As known,

Equation of a line:$\frac{x}{a} +\frac{y}{b} =1$

Where $a=x-intercept\ b=\ y-intercept$

Given that line intersects $y-axis$ at P

P lies on $y-axis$ and $p=( 0,\ b)$

Line intersects $x-axis$ at $Q$

$Q$ lies on $x-axis$ and $Q=( a,\ 0)$

On using the mid point formula.

$( x,\ y) =\left(\frac{x_{1} +x_{2}}{2} ,\ \frac{y_{1} +y_{2}}{2}\right)$

Midpoint of $PQ=\left(\frac{a+0}{2} ,\ \frac{0+b}{2}\right) =\left(\frac{a}{2} ,\frac{b}{2}\right)$

$\because$ Mid point given $( 2,\ -5)$

$\left(\frac{a}{2} , \frac{b}{2}\right) =( 2,\ -5)$

$\Rightarrow \frac{a}{2} =2$ and $\frac{b}{2} =-5$

$\Rightarrow a=4$ and $b=-10$

Thus $P=( 0,\ -10)$ and $Q=( 4,\ 0)$.