The line 3x-4y=12 is tangent to the parabola y^2=4px find p

Question

anonymous · Answer

@SolomonZelman

solomonzelman · Answer

3x-4y=12        --->        -4y=-3x+12        --->        y = (3/4)x - 3

solomonzelman · Answer

So the slope of your parabola at the point of tangency is 3/4.

anonymous · Answer

ok

solomonzelman · Answer

y^2 = 4px

 y = ±$\sqrt{4px}$

 y = ±2$\sqrt{px}$

 y$'$ = ±p$/\sqrt{px}$

Where is the slope 3/4?

3/4 = ±p$/\sqrt{px}$

and it is above the x axis because the slope is positive, so in ± we take the +.

3/4 = p$/\sqrt{px}$
3/(4p) = $1/\sqrt{px}$
(4p)/3 = $\sqrt{px}$
4p = 3√(px)        -----> first equation

solomonzelman · Answer

4p = 3√(px)
4p = 3•√p•√x
4√p = 3√x
16p = 9x
16p / 9 = x  ---> point of tangency

solomonzelman · Answer

oh sorry we choose the below the x-axis part. SO http://www.wolframalpha.com/input/?i=%283%2F4%29%2816p+%2F+9%29+-+3+%3D+-2%E2%88%9A%28p%2816p+%2F+9%29%29

anonymous · Answer

so what do we do with the point of tangency

solomonzelman · Answer

https://www.desmos.com/calculator/bj3xaweyae

solomonzelman · Answer

I solved for p via wolfram and there we see it is once tangent to the p=-9/4 version parabola.