For which values of a and b is the line b 2x + y = tangent to the parabola 2
y = ax
when x = 2? (2)
y=ax2y′=2axy{at x=2}′=Gradient=2(2)(a)=4aAt x=2,y=4aEquation of tangenty−4a=4a(x−2)y=4ax+4a−8a=4a(x−1)If the line is equal with 2x+y=b, we can compare2x+y=by−4ax=−4a2=−4a,a=−12, b=−4a=−4×−12=2∴a=−12, and b=2\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 = 2y=ax2y′=2axy{atx=2}′=Gradient=2(2)(a)=4aAtx=2,y=4aEquation of tangenty−4a=4a(x−2)y=4ax+4a−8a=4a(x−1)If the line is equal with2x+y=b,we can compare2x+y=by−4ax=−4a2=−4a,a=−21,b=−4a=−4×−21=2∴a=−21,andb=2
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