1.29: Perpendicular Lines and a Quadrilateral

4PM1/1/January/2019 — Question 9 · 16 marks

The point A\displaystyle A has coordinates (−3,−6)\displaystyle (-3,-6) and the point B\displaystyle B has coordinates (5,−2)\displaystyle (5,-2).
The line l\displaystyle l passes through the point A\displaystyle A and the point B\displaystyle B.
(a) Find an equation of l\displaystyle l, giving your answer in the form
y=mx+c.y=mx+c.
(3)
The point P\displaystyle P has coordinates (k,−2)\displaystyle (k,-2). The line through A\displaystyle A and P\displaystyle P is perpendicular to l\displaystyle l.
(b) Show that k=−5\displaystyle k=-5.
(3)
The point Q\displaystyle Q has coordinates (e,f)\displaystyle (e,f). The line through B\displaystyle B and Q\displaystyle Q is also perpendicular to l\displaystyle l.
Given that the length of PQ\displaystyle PQ is 85\displaystyle \sqrt{85} and that f>0\displaystyle f>0,
(c) find the coordinates of Q\displaystyle Q.
(6)
(d) Calculate the area of quadrilateral ABQP\displaystyle ABQP.
(4)