Question

The line x - 4y=6 is the perpendicular bisector of the line segment AB. If B=(1,3); find the coordinates of point A.

Answer 1

hola there....put PQ as AB , B=P and Q =A

hope you like it

Answer 2

The perpendicular line of x - 4y = 6 will be
4x + y = k

As the line 4x + y = k is made by joining point
A(p,q) and point B(1,3)

therefore,

4x + y = k
4(1) + (3) = k
k = 7

So line AB is 4x + y = 7
So now we got that the line x - 4y = 6 is perpendicular bisector of 4x + y = 7

As line x - 4y = 6 is perpendicular bisector it will meet line AB at it's midpoint.

So intersection of both the lines will be i.e. of
4x + y = 7 and x - 4y = 6 is (2,-1)

so midpoint of line AB is (2,-1)

By midpoint formula,

$(2, - 1) = ( \frac{p + 1}{2} , \frac{q + 3}{2} )$

So,

$2 = \frac{p + 1}{2}$

$p = 3$

Similarly,

$- 1 = \frac{q + 3}{2}$

$q = - 5$

So point A is (3,-5)