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
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:
$\displaystyle y + 2x = k \Rightarrow y = k - 2x$ .... (1)
$\displaystyle y = x^2 - 6x + 14$ .... (2)
Start solving equations (1) and (2) simultaneously:
$\displaystyle k - 2x = x^2 - 6x + 14 \Rightarrow x^2 - 4x + (14 + k) = 0$.
There will be only one solution (and hence $\displaystyle y + 2x = k$ will be a tangent to $\displaystyle y = x^2 - 6x + 14$) if the discriminant of this quadratic is equal to zero:
$\displaystyle (-4)^2 - 4(1)(14 + k) = 0 \Rightarrow k = \, ....$