Question

A line meets the straight lines 5x − y − 4 = 0 and 3x + 4y − 4 = 0 at points P and Q. If (1, 5) is the midpoint of PQ, find the equation of the line PQ,

Answer 1

 See Fig. Let the line PQ be

y - 5 = m(x - 1)     ... (1)  

 Substituting y = mx + 5 - m in the equation 5x - y - 4 = 0,

we have

5x - mx - 5 + m -4 = 0

(5 -m)x + m - 9 = 0

Therefore

 Substituting y = mx + 5 - m  in the equation 3x + 4y - 4 = 0,

we have

3x + 4(mx + 5 - m) - 4 = 0

(3 + 4m)x + 16 - 4m = 0

From Eq.(3), We get

Substituting the value of m  = 83/35 in Eq. (1), equation of line PQ is obtained as

y - 5 = 83/35(x - 1)

⇒ 83x - 35y + 92 = 0

The value of m obtained from Eq. (3) is also equal to 83/35.