Question

Point P(0,2) is the point of intersection of y axis and perpendicular bisector of line segment joining points A(-1,1) and B(3,3).State true or false and give reason.

Answer 1

we know that the slope of the line passing through the points $\left(x_1,y_1\right)\mathrm{and}\left(x_2,y_2\right)$ is $\frac{y_2-y_1}{x_2-x_1}$.
So the slope of the line passing through the points A(-1, 1) and B(3, 3) = $\frac{3-1}{3-\left(-1\right)}=\frac{2}{4}=\frac{1}{2}$
And the midpoint of the points A(-1, 1) and B(3, 3) is P $\left(\frac{-1+3}{2},\frac{1+3}{2}\right)\mathrm{i}.\mathrm{e}.\left(1,2\right)$.
Since the slope of line connecting points A and B is $\frac{1}{2}$.
Then, slope of the perpendicular bisector through point P will have slope as; $\frac{-1}{\mathrm{m}}=\frac{-1}{{\displaystyle \frac{1}{2}}}=-2$.
So the equation of the line segment of perpendicular bisector is $y=-2x+c$.
And since this perpendicular bisector passes through the point P(1, 2) , so we have;
$$\begin{gathered} 2=-2+c \\ \Rightarrow c=4 \end{gathered}$$
Therefore the equation of perpendicular bisector is $y=-2x+4$.
Now putting x = 0 we get;
$y=0+4=4$
So the point (0, 2) does not lie on the line $y=-2x+4$.
Therefore the point (0, 2) is not the point of intersection of y axis and perpendicular bisector of line segment joining points A(-1,1) and B(3,3).
Hence the statement is false.