Question 1032140: Consider the parabola y = f(x) = x^2 − 2x and a point M(a, b) on the parabola.
(a) Find b in terms of a. Then find the equation of the tangent to the parabola at point M (in terms of a).
(b) Find the equations of the two tangents to the parabola which pass through the point P(0, −4).
(Note that point P(0, −4) is NOT on the parabola).
(c) The two tangents in part (b) contact the parabola at points C and D. Find the area of triangle PCD.
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! (a)  ==>  ==> 
(b) The derivative is y' = f'(x) = 2x - 2. ==> f'(a) = 2a - 2 gives the slope of the tangent line to the parabola at the point (a,b).
==> the equation of the tangent line at (a,b) is y - b = (2a-2)(x-a).
Since the line is also supposed to pass through (0,-4), we get after substitution,
-4 - b = (2a-2)(0-a), or 4+b = a(2a-2)
==> 
==> 4+b = 2b+2a ==> b + 2a = 4 <----Equation A.
Solving Equation A simultaneously with the equation , we get
 ==>  , or a = -2 or a = 2.
The corresponding b values are b = 8 or b = 0.
Hence the two contact points are (-2,8) and (2,0), and the two tangent lines are
y - 8 = -6(x+2), and
y = 2(x - 2).
(c) Area =  square units.
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