Chapter 11, Problem 2RCC

a.

To determine

To define: Geometrical definition and foci of ellipse.

Answer to Problem 2RCC

Foci or focal points are the two points inside the ellipse. These points lie on major axis spaced equally each side of the center.

Explanation of Solution

Given information:

The geometric definition of ellipse.

Concept Used:

An ellipse is a plane curve surrounding two focal points, such that for all points on the curve, sum of two distances to the focal point is constant. In the figure 1 $F_1$ and $F_2$ are focal points.

Calculation:

From definition of ellipse

  

Conclusion:

The foci of ellipse are

  $C_1+C_2=C_3+C_4$

b.

To determine

Coordinate of the foci minor and major axes major axes of ellipse.

Explanation of Solution

Given information:

Equation of ellipse

  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

where $a>b>0$

Graph :

  

Interpretation :

  $F_1$ and $F_2$ are focal points.

Coordinate of vertices are $\left(\pm a,0\right)$

Coordinate of co − verticies are $\left(0,\pm b\right)$

Coordinate of focal points are $\left(\pm c,0\right)$ where $c^2=a^2-b^2$ .

Since $a>b$ , hence major axis is parallel to x − axis.

c.

To determine

Eccentricity of ellipse

Answer to Problem 2RCC

The eccentricity $=\frac{c}{a}$

Explanation of Solution

Given information:

Given information:

Equation of ellipse

  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

where $a>b>0$

Concept Used:

Eccentricity of ellipse is the ratio of the distance from center to vertices. For given equation of ellipse.

Calculation:

Distance between foci to centre $=c$

Distance between center to vertices $=a$

Conclusion:

Hence eccentricity $=\frac{c}{a}$

d.

To determine

Equation of ellipse with foci on y - axis

Explanation of Solution

Given information:

Equation of ellipse

  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

where $a>b>0$

Concept Used:

In this case major axis is parallel to y − axis.

Calculation:

In this case major axis is parallel to y − axis. Hence equation of ellipse

  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Where $a>b$

Conclusion:

The required equation of ellipse

  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Where $a>b$