# 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 $Q=( 4,\ 0)$.

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