Parabolas are used to represent curved functions
The shortest distance between the parabola and the point is [tex]x = -\frac 1{2}[/tex]
How to determine the shortest distance
The parabola is given as:
[tex]y=(7x+5)^{1/2}[/tex]
Rewrite as:
[tex]y=\sqrt{7x+5}[/tex]
Let p represent a point on the parabola.
So, we have:
[tex]P(x,\sqrt{7x + 5})[/tex]
Next, calculate the distance between P and the point (3,0) using the following distance formula:
[tex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2[/tex]
So, we have:
[tex]d = \sqrt{(x- 3)^2 + (\sqrt{7x + 5})^2[/tex]
[tex]d = \sqrt{x^2- 6x + 9 + 7x + 5}[/tex]
Evaluate the like terms
[tex]d = \sqrt{x^2+ x + 14}[/tex]
Square both sides
[tex]d^2 = x^2+ x + 14[/tex]
The shortest distance is then calculated as:
[tex]x = -\frac b{2a}[/tex]
Where:
a = 1, b = 1 and c = 14
So, we have:
[tex]x = -\frac 1{2*1}[/tex]
[tex]x = -\frac 1{2}[/tex]
Hence, the shortest distance between the parabola and the point is [tex]x = -\frac 1{2}[/tex]
Read more about parabolas at:
https://brainly.com/question/4061870
