Question

the line segment joining P(-4,5) and Q(3,2) intersect the y- axis at R. PM and QN are perpendiculars from P and Q on the x axis.
find: 1) the ratio PR:QR
2) the cordinates of R
3) the area of the quadrilateral PMNQ

Answer 1

Slope of PQ = (2 – 5)/(3 + 4) = –3/7.
Equation of PQ is y – 5 = – 3/7(x + 4)
Or, 7y – 35 = – 3x – 12
Or, 3x + 7y – 23 = 0 ---------------------- (i)
This passes through y – axis , putting x = 0 in (i) we get,
7y = 23 => y = 23/7 ;
the co-ordinates of R is (0, 23/7).

1. Let the ratio be k : 1
   Therefore, 0 = {k×3 + 1×(– 4)}/(k + 1)
   Or, 3k – 4 = 0 => k = 4/3 => k : 1 = 4 : 3
   Or, PR : RQ = 4 : 3. [Ans.]
   
2. Co-ordinates of R is R(0, 23/7). [Ans.]
   
3. Area of quadrilateral PMNQ = 1/2×[PM + QN]×MN
   = 1/2×(5 + 2)×7
   = 1/2×7×7
   = 49/2 = 24.5 sq. units. [Ans.]