For what values of a and b is the line 2x+y=b tangent to the parabola y=ax^2 when x=6?

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The gradient of the tangent to a curve at any point is given by its derivative.

`y = ax^2`

`dy/dx = 2ax`

The gradient of the tangent to the above curve at x = 6 is given by `(dy/dx)_(x = 6)`

`(dy/dx)_(x = 6) = 2xxaxx6 = 12a`

It is given that 2x+y = b is tangent to the above curve at x = 6.

`2x+y = b`

`y = -2x+b`

Since the gradients are equal for the tangent;

`-2 = 12a`

`a = -6`

`y = ax^2`

`y = -6x^2`

when x = 6 then y = -256

Now we know that tangent point is (-6,-256). This tangent point lies on the line 2x+y=b also.

`2xx6-256 = b`

`b = -244`

So the answers are;

a = -6

b = -244

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