Answer to Question #163445 in Calculus for Happy

Question

For which values of a and b is the line b 2x + y = tangent to the parabola 2

y = ax

when x = 2? (2) 

 

Answer 1

$\displaystyle y = ax^2 \\ y' = 2ax \\ y'_{\{\textsf{at}\,\, x = 2\}} = \textsf{Gradient} = 2(2)(a) = 4a \\ \textsf{At}\,\,\, x = 2, y = 4a \\ \textsf{Equation of tangent}\\ y - 4a = 4a(x - 2)\\ y = 4ax + 4a - 8a = 4a(x - 1)\\ \textsf{If the line is equal with}\,\,2x + y = b,\,\,\textsf{we can compare}\\ 2x + y = b \\ y - 4ax = -4a\\ 2 = -4a, a = -\frac{1}{2},\,\, b = -4a = -4 \times -\frac{1}{2} = 2\\ \therefore a = -\frac{1}{2},\,\,\textsf{and}\,\, b = 2$