12 The diagram shows part of curve with equation = 8x + 20 and part of the line with y x2 – equation = + 6. y x (a) Using an appropriate algebraic method, find the coordinates of and A B. (4) The of and are denoted and respectively. x-coordinates A B xA xB (b) Find the exact value of the area of the finite region bounded by the the lines x-axis, = and = and the line x xA x xB AB. (2) (c) Use calculus to find the exact value of the area of the finite region bounded by the x-axis, the lines = and = and the curve = 8x + 20. x xA x xB y x2 – (5) (d) Hence, find, to one decimal place, the area of the shaded region enclosed by the curve = 8x + 20 and the line y x2 AB. – (2) (Total 13 marks) ___________________________________________________________________________ is the centre of circle with equation 8x + + 10y + 1 = 0. 13. A C, x2 y2 – and are points on the circle and the lines and are tangents to the circle at these P, Q R l1, l2 l3 points respectively. Line intersects line at and line at l2 l1 B l3 D. l2 l1 y P B E x Q l3 D
12 The diagram shows part of curve with equation y = x2 – 8x + 20 and part of the line with equation y = x + 6. (a) Using an appropriate algebraic method, find the coordinates of A and B. (4) The x-coordinates of A and B are denoted xA and xB respectively. (b) Find the exact value of the area of the finite region bounded by the x-axis, the lines x = xA and x = xB and the line AB. (2) (c) Use calculus to find the exact value of the area of the finite region bounded by the x-axis, the lines x = xA and x = xB and the curve y = x2 – 8x + 20. (5) (d) Hence, find, to one decimal place, the area of the shaded region enclosed by the curve y = x2 – 8x + 20 and the line AB. (2) (Total 13 marks) ___________________________________________________________________________ 13. A is the centre of circle C, with equation x2 – 8x + y2 + 10y + 1 = 0. P, Q and R are points on the circle and the lines l1, l2 and l3 are tangents to the circle at these points respectively. Line l2 intersects line l1 at B and line l3 at D. l2 l1 y P B E x Q l3 D