Differentiation Tangents and Normals

Question

Find the equation of the tangent to the curve y=x+x^2 at the point where x=a. Find the values of a for which this line passes through the point P(2,-3).Hence, find the equations of the tangents from P to the curve.

Answer 1

y = x + x²

 

y' = 1 + 2x = slope of tangent line at (x, x+x²)

 

Slope of tangent line at x = a:  1 + 2a

 

Equation of tangent line at (a, a + a²):

 

    y - (a + a²) = (1 + 2a)(x - a)

 

   If (2, -3) lies on the tangent line, then -3 - (a+a²) = (1+2a)(2-a)

 

  -3 - a - a² = -2a² + 3a + 2

 

   a² -4a - 5 = 0

 

  (a - 5)(a + 1) = 0       So, a = 5, -1