parabola of a tangent

label Mathematics
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For what values of a and b is the line

4x + y = b
tangent to the parabola
y = ax2
when x = 4?

Jul 15th, 2015

Thank you for the opportunity to help you with your question!

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Differentiating y w.r.t. x


This is the slope of tangent to the parabola at any point x.


At x = 4

Slope of the tangent is



Given that 

4x + y = b 

is a tangent to the parabola at x = 4


The equation of tangent can be rewritten as

y = -4x + b

Comparing the above equation with slope intercept form of a line

y = mx + b (where m is the slope of the line)

we get, slope of the tangent = -4


Therefore, 



So, equation of parabola becomes



At x = 4

y = 


(4, -8) is the point at which the tangent touches the parabola.

Substitute x = 4 and y = -8 in the equation of tangent to find b



ANSWER:  

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Please let me know if you need any clarification. I'm always happy to answer your questions.
Jul 15th, 2015

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