Question

If the line x - 4y - 6 = 0 is the perpendicular bisector of the line segment PQ and the coordinates of P are (1,3), find the coordinates of Q.

Answer 1

Consider the perpendicular bisector x−4y−6=0
⇒4y=x−6
⇒y=14x−64......(i)

⇒Slope of perpendicular bisector=14
The line x−4y−6=0 is the perpendicular bisector of the line segment PQ.
⇒Slope of PQ=−4
The equation of the line PQ passing through P (1, 3) having slope −4 is given by:(y−3)=−4(x−1)
⇒y−3=−4x+4
⇒y=−4x+7......(ii)
The perpendicular bisector x−4y−6=0 intersect PQ at its mid point.
The mid point is given by substituting (ii) in (i):−4x+7=1/4x−6/4⇒7+6/4=1/4x+4x
⇒34/4=17/4x
⇒x=2 and y=−1
Therefore, the mid point of PQ is (2, −1).Let the coordinates of Q be (x0, y0) and we know that
P is (1, 3).
Mid point of PQ
=(1+x0)/2, (3+y0)/2)
⇒(2, −1)=((1+x0)/2, (3+y0)/2)
⇒(1+x0)/2=2⇒1+x0=4
⇒x0=3and (3+y0)/2=−1
⇒3+y0=−2
⇒y0=−5
Thus, the coordinates of Q are (3, −5).