# Thread: Tangent to a curve; constant k

1. ## Tangent to a curve; constant k

Hi guys,

I am confused with this one. Would I still use dy/dx? If not, could someone please provide me with a formula?

Find the value of the constant k for which the line y + 2x = k is a tangent to the curve y = x ² - 6x + 14

Thanks

2. Originally Posted by laoch
Hi guys,

I am confused with this one. Would I still use dy/dx? If not, could someone please provide me with a formula?

Find the value of the constant k for which the line y + 2x = k is a tangent to the curve y = x ² - 6x + 14

Thanks
The simplest approach is to solve for the intersection points of the line and parabola and then force the value of k to be such that there's only one intersection point:

$y + 2x = k \Rightarrow y = k - 2x$ .... (1)

$y = x^2 - 6x + 14$ .... (2)

Start solving equations (1) and (2) simultaneously:

$k - 2x = x^2 - 6x + 14 \Rightarrow x^2 - 4x + (14 + k) = 0$.

There will be only one solution (and hence $y + 2x = k$ will be a tangent to $y = x^2 - 6x + 14$) if the discriminant of this quadratic is equal to zero:

$(-4)^2 - 4(1)(14 + k) = 0 \Rightarrow k = \, ....$

3. Thank you so much