MHB Finding the equation of a curve given the tangent equation

[musad]

I have a calculus question and I am not sure where to get started. The question states:"Find the value of c>0 such that the line y=x+1 is the tangent line to the curve c√x (i.e. intersects the curve at one point and shares the same slope at that point)."
Thanks

 

[Prove It]

I have a calculus question and I am not sure where to get started. The question states:"Find the value of c>0 such that the line y=x+1 is the tangent line to the curve c√x (i.e. intersects the curve at one point and shares the same slope at that point)."
Thanks

$\displaystyle \begin{align*} x + 1 &= c\,\sqrt{x} \\ \left( x +1 \right) ^2 &= \left( c\,\sqrt{x} \right) ^2 \\ x^2 + 2x +1 &= c^2\,x \\ x^2 + \left( 2 - c^2 \right) \, x + 1 &= 0 \end{align*}$

Now to have only one solution, the discriminant must be zero...

 

[musad]

What is the discriminant?

 

[Prove It]

What is the discriminant?

You're doing calculus without knowledge of basic quadratics?

 

[musad]

You're doing calculus without knowledge of basic quadratics?

I do, I just think we may use different terminology so i wasnt sure what the term meant, sorry.

 

[Prove It]

I do, I just think we may use different terminology so i wasnt sure what the term meant, sorry.

For a quadratic of the form $\displaystyle \begin{align*} a\,x^2 + b\,x + c = 0 \end{align*}$, the roots are $\displaystyle \begin{align*} x = \frac{-b \pm \sqrt{b^2 - 4\,a\,c}}{2a} \end{align*}$. The stuff under the square root is important, because it determines whether or not there are solutions (as you can't have the square root of a negative number). It's so important, it has its own name, the DISCRIMINANT.