# A line intersects the y-axis and x-axis at the points P and Q respectively. If $( 2,\ -5)$ is the mid-point then find the coordinates of P and Q.

Given: A line intersects the y-axis and x-axis at the points P and Q respectively. $( 2,\ -5)$ is the mid-point.

To do: 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 $Q=( 2,\ 0)$

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Updated on: 10-Oct-2022

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