The value of [tex]c[/tex] such that the line [tex]y = 2\cdot x + 3[/tex] is tangent to the parabola [tex]y = c\cdot x^{2}[/tex] is [tex]-\frac{1}{3}[/tex].
If [tex]y = 2\cdot x + 3[/tex] is a line tangent to the parabola [tex]y = c\cdot x^{2}[/tex], then we must observe the following condition, that is, the slope of the line is equal to the first derivative of the parabola:
[tex]2\cdot c \cdot x = 2[/tex] (1)
Then, we have the following system of equations:
[tex]y = 2\cdot x + 3[/tex] (1)
[tex]y = c\cdot x^{2}[/tex] (2)
[tex]c\cdot x = 1[/tex] (3)
Whose solution is shown below:
By (3):
[tex]c =\frac{1}{x}[/tex]
(3) in (2):
[tex]y = x[/tex] (4)
(4) in (1):
[tex]y = -3[/tex]
[tex]x = -3[/tex]
[tex]c = -\frac{1}{3}[/tex]
The value of [tex]c[/tex] such that the line [tex]y = 2\cdot x + 3[/tex] is tangent to the parabola [tex]y = c\cdot x^{2}[/tex] is [tex]-\frac{1}{3}[/tex].
We kindly invite to check this question on tangent lines: https://brainly.com/question/13424370
