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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