Hunterfreddyidn
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A parabola is defined as y=(7x+5)^1/2. A point (3,0) is located near the parabola. Show that x=1/2 to get the shortest distance between the parabola and the point.​

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MrRoyalidn

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

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