Question

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.​

Answer 1

Parabolas are used to represent curved functions

The shortest distance between the parabola and the point is $x = -\frac 1{2}$

How to determine the shortest distance

The parabola is given as:

$y=(7x+5)^{1/2}$

Rewrite as:

$y=\sqrt{7x+5}$

Let p represent a point on the parabola.

So, we have:

$P(x,\sqrt{7x + 5})$

Next, calculate the distance between P and the point (3,0) using the following distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2$

So, we have:

$d = \sqrt{(x- 3)^2 + (\sqrt{7x + 5})^2$

$d = \sqrt{x^2- 6x + 9 + 7x + 5}$

Evaluate the like terms

$d = \sqrt{x^2+ x + 14}$

Square both sides

$d^2 = x^2+ x + 14$

The shortest distance is then calculated as:

$x = -\frac b{2a}$

Where:

a = 1, b = 1 and c = 14

So, we have:

$x = -\frac 1{2*1}$

$x = -\frac 1{2}$

Hence, the shortest distance between the parabola and the point is $x = -\frac 1{2}$

Read more about parabolas at:

https://brainly.com/question/4061870