Question

I will Fan and Medal the best answer.

At what points on y=3x^3+14x^2+3x+8 does the tangent pass through the origin.

Answer 1

If we have two points (x1, y1) and (x2, y2) on a line, the slope of this line is defined as: (y2 - y1) /( x2 - x1) = m. If this is a line tangent to the curve we must have one of the points (x2, y2) lie on the curve and m = (derivative of curve at that point). Also, since the other point passes through the origin we can take (x1, y1) to be (0,0). Written out: y/x = dy/dx (m = dy/dx and I removed the 2's). Just solve this equation for x.

Answer 2

Thanks @andrewyates. I got -3 and -1/9 as the answer. Is this is correct ??

Answer 3

@andrewyates You googled this

Answer 4

FInd dy / dx to get an expression for the slope of a tangent to the curve

dy/dx = 9x^2 + 28x + 3

Answer 5

now the tangents to the curve go through (0,0) so the equations of tangent is y = mx

y = 3x^3 + 14x^2 + 2x + 8
y = mx = 9x^3 + 28x^2 + 3x

so we have
3x^3 + 14x^2 + 2x + 8 = 9x^3 + 28x^2 + 3x

solving this for x will give the x coordinate of the points of contact of the tangent
then you can find the value of m

I drew the curve on my calculator and it looks like there are 2 tangents that go through (0,0)

Answer 6

the equation simplifies to
6x^3 + 14x^2 - 8 = 0

which is not easy to solve

I would use a graphical calculator if you have one or the online graphical calculator Desmos.

! root is x = -1 so m = 9(-1)^2 + 28(1) + 3 = -16

so one tangent is y = -16x and this passes through th point (-1,16) . So thats one answer.

Answer 7

Do you understand how I got the equation

3x^3 + 14x^2 + 2x + 8 = 9x^3 + 28x^2 + 3x?

Answer 8

yes i understand. Thanks for the help @welshfella

Answer 9

yw