Question

What is the value of c such that the line y=2x+3 is tangent to the parabola y=cx^2

Answer 1

The value of $c$ such that the line $y = 2\cdot x + 3$ is tangent to the parabola $y = c\cdot x^{2}$ is $-\frac{1}{3}$.

If $y = 2\cdot x + 3$ is a line tangent to the parabola $y = c\cdot x^{2}$, then we must observe the following condition, that is, the slope of the line is equal to the first derivative of the parabola:

$2\cdot c \cdot x = 2$ (1)

Then, we have the following system of equations:

$y = 2\cdot x + 3$ (1)

$y = c\cdot x^{2}$ (2)

$c\cdot x = 1$ (3)

Whose solution is shown below:

By (3):

$c =\frac{1}{x}$

(3) in (2):

$y = x$ (4)

(4) in (1):

$y = -3$

$x = -3$

$c = -\frac{1}{3}$

The value of $c$ such that the line $y = 2\cdot x + 3$ is tangent to the parabola $y = c\cdot x^{2}$ is $-\frac{1}{3}$.

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