1.15: Figure 1 shows part of the curve with equation where , and are constants.

4PM1/2/November/2023 — Question 5 · 15 marks

1.15 diagram 1
Figure 1 shows part of the curve S\displaystyle S with equation y=px2+qx+r\displaystyle y = px^2 + qx + r where p\displaystyle p, q\displaystyle q and r\displaystyle r are constants.
The points A\displaystyle A, B\displaystyle B and P\displaystyle P with coordinates (−2,0)\displaystyle (-2, 0), (6,0)\displaystyle (6, 0) and (4,−6)\displaystyle (4, -6) respectively lie on S\displaystyle S
(a) Show that an equation of S\displaystyle S is y=x22−2x−6\displaystyle y = \frac{x^2}{2} - 2x - 6
(3)
The line l\displaystyle l is the normal to S\displaystyle S at the point P\displaystyle P
(b) Show that an equation of l\displaystyle l is 2y+x+8=0\displaystyle 2y + x + 8 = 0
(5)
The finite region shown shaded in Figure 1 is bounded by S\displaystyle S and l\displaystyle l
(c) Use algebraic integration to find the exact area of the shaded region.
(7)