Differentiation Tangents and Normals

Question

Find the coordinates of the two points on the curve y= 4- x^2 whose tangents passes through the point (-1 ,7 ).

Answer 1

y = 4 - x²

 

Slope of tangent line at (x, 4-x²) = -2x

 

Slope of line through (x, 4-x²) and (-1, 7) =

 

                   [7 - (4-x²)]/[-1-x] = -2x

 

                   So, 7 - 4 + x² = 2x + 2x²

 

                         x² + 2x - 3 = 0

 

                        (x + 3)(x - 1) = 0      x = -3, 1

 

When x = -3, y = 4 - x² = -5

When x = 1, y = 4 - x² = 3

 

Points on the curve y = 4 - x² whose tangents pass through the point (-1, 7):

 

     (-3, -5) and (1, 3)