Question

The line 𝑦=𝑥−5 is tangent to the curve 𝑦=𝑥^2−9𝑥−𝑘

Find the value of 𝑘.

Answer 1

The slope of the tangnet line $y=x-5$ is $m=1$.

$y=x^2-9x-k$ , $\frac{dy}{dx} =2x-9=1$

We have $2x-9=1$ ,

$2x=10$ ,

$x=5$.

Then $y=x-5=0$.

So, The line $y=x-5$ is tangent to the curve $y=x^2-9x-k$ at the point $(5, \ 0)$ .

$0=5^2-9\cdot 5-k$

$k=-20$

Answer: $k=-20$