derivative application

**slaypullingcat** · April 25th 2009, 10:32 PM

Find the equation of any quadratics that passes through the origin and are tangent to both y=-2x-4 and y=8x-49.

I have thought about this problem for most of the afternoon now and got no where. I tried drawing it to see the problem but no good. I just don't know where to start or how to approach this problem. What/how should I be thinking?

Thank you

**running-gag** · April 26th 2009, 12:57 AM

Hi

The quadratics you are looking for passing through the origin, their generic equation is $f(x) = ax^2 + bx$

The tangent at a point whose abscissa is $x_0$ is :
$T_0 : y = (2ax_0+b)x-ax_0^2$

The quadratics are tangent to both y=-2x-4 and y=8x-49 iff there exists $x_0$ and $x_1$ such that
$2ax_0+b = -2$
$-ax_0^2 = -4$
$2ax_1+b = 8$
$-ax_1^2 = -49$

Solve for a, b, x0 and x1.
You will find 2 quadratics.

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