Chapter 10, Problem 75RE

To determine

Tofind:the collection of the points in a plane if distance from the point is three- fourths of its distance from the line.

Answer to Problem 75RE

The collection all point give an ellipses geometry and the equation of the ellipse is $\frac{x^2}{16}+\frac{y^2}{7}=0$ .

Explanation of Solution

Given:

Distance from the point $\left(\text{3,0}\right)$ .

  ${\text{d}}_{\text{1}}\text{=}\frac{\text{3}}{\text{4}}{\text{d}}_{\text{2}}$ .

Distance from the line $\text{x=}\frac{\text{16}}{\text{3}}$ .

Concept used:

Distance formula:

  $\text{d=}\sqrt{{\left(\text{x-h}\right)}^{\text{2}}\text{+}{\left(\text{y-k}\right)}^{\text{2}}}$ .

Calculation:

Let any point on the curve be $\left(\text{x,y}\right)$

Distance from the point $\left(\text{3,0}\right)$ .

  ${\text{d}}_1=\sqrt{{\left(x-3\right)}^2+y^2}$ .

Distance from the line $\text{x=}\frac{\text{16}}{\text{3}}$ .

  ${\text{d}}_{\text{2}}\text{=x-}\frac{\text{16}}{\text{3}}$ .

  ${\text{d}}_{\text{1}}\text{=}\frac{\text{3}}{\text{4}}{\text{d}}_{\text{2}}$ .

  $\frac{\text{3}}{\text{4}}\left(\text{x-}\frac{\text{16}}{\text{3}}\right)\text{=}\sqrt{{\left(\text{x-3}\right)}^{\text{2}}{\text{+y}}^{\text{2}}}$ .

  $\begin{array}{l}\frac{9}{16}\left({\text{x}}^2+\frac{\text{16}}{9}\right)\text{=}\frac{9}{16}{\left(\text{x-3}\right)}^{\text{2}}{\text{+y}}^{\text{2}}\\ \end{array}$ .

  $\frac{7x^2}{16}+y^2=7$ .

  $\frac{x^2}{16}+\frac{y^2}{7}=0$ .

Hence, the collection all point give an ellipses geometry and the equation of the ellipse is $\frac{x^2}{16}+\frac{y^2}{7}=0$ .