Question from http://www.sgforums.com/forums/2297/topics/309076
Find the value of the constant c for which the line y = 2x + c is a tangent to the curve y = 4x² - 6x + 11. This tangent meets the x-axis at A and the y-axis at B. Calculate the area of the triangle AOB, where O is the origin.
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Answer:
Coordinate Geometry Method
4x² - 6x + 11 = 2x + c
4x² - 8x + 11 - c = 0
For roots to be real and equal (tangent to curve), b² - 4ac = 0
64 - 4(4)(11-c) = 0
16c = 112
c = 7
Differentiation Method
y=4x² - 6x + 11
⇒ dy/dx = 8x - 6
for the tangent of the curve is // to y = 2x + c,
⇒ 8x - 6 = 2
⇒ x = 1
⇒ y = 4 - 6 + 11 = 9
⇒ the tangent meets the curve at (1,9)
since (1,9) is a point on y = 2x + c,
⇒ 9 = 2 + c
⇒ c = 7
When y = 2x + 7 meets the x-axis, y=0, i.e.
2x + 7 = 0
x = -3½
⇒ A is (-3½,0).
When y = 2x + 7 meets the y-axis, x=0, i.e.
y = 2(0) + 7 = 7
⇒ B is (0,7).
Area of ∆AOB =
= ½ | -24½ |
= 12¼ units²
