Chapter 9, Problem 156RE

a.

To determine

To find: The variation caused by the equation $\frac{x^2}{9}+\frac{y^2}{4}=1$ in the original equation $\frac{x^2}{4}+\frac{y^2}{9}=1$ .

Answer to Problem 156RE

The equation $\frac{x^2}{9}+\frac{y^2}{4}=1$ , rotates the graph of the original equation by an angle $\frac{\pi }{2}$ .

Explanation of Solution

Given:

The original equation $\frac{x^2}{4}+\frac{y^2}{9}=1$ and variation $\frac{x^2}{9}+\frac{y^2}{4}=1$ .

Calculation:

The original equation $\frac{x^2}{4}+\frac{y^2}{9}=1$ represents a vertical ellipse having the length of major axis as 6 units and length of minor axis as 4 units.

The equation $\frac{x^2}{9}+\frac{y^2}{4}=1$ , represents a horizontal ellipse having length of major axis as 6 and minor axis as 4. That means, the equation $\frac{x^2}{9}+\frac{y^2}{4}=1$ , rotates the graph of the original equation by an angle $\frac{\pi }{2}$ .

b.

To determine

To find: The variation caused by the equation $\frac{x^2}{4}+\frac{y^2}{4}=1$ in the original equation $\frac{x^2}{4}+\frac{y^2}{9}=1$ .

Answer to Problem 156RE

The length of the major axis is reduced by 2 units to form a circle of radius 2 units

Explanation of Solution

Given:

The original equation $\frac{x^2}{4}+\frac{y^2}{9}=1$ and variation $\frac{x^2}{4}+\frac{y^2}{4}=1$ .

Calculation:

The original equation $\frac{x^2}{4}+\frac{y^2}{9}=1$ represents a vertical ellipse having the length of major axis as 6 units and length of minor axis as 4 units.

For the equation $\frac{x^2}{4}+\frac{y^2}{4}=1$ , $a=b=2$ . So, the equation can be written as $x^2+y^2=4$ , which is an equation of a circle with radius 2.

That means, the length of the major axis is reduced by 2 units to form a circle of radius 2 units.

c.

To determine

To find: The variation caused by the equation $\frac{x^2}{4}+\frac{y^2}{25}=1$ in the original equation $\frac{x^2}{4}+\frac{y^2}{9}=1$ .

Answer to Problem 156RE

The graph of the original equation is elongated vertically.

Explanation of Solution

Given:

The original equation $\frac{x^2}{4}+\frac{y^2}{9}=1$ and variation $\frac{x^2}{4}+\frac{y^2}{25}=1$ .

Calculation:

The original equation $\frac{x^2}{4}+\frac{y^2}{9}=1$ represents a vertical ellipse having the length of major axis as 6 units and length of minor axis as 4 units.

The new equation $\frac{x^2}{4}+\frac{y^2}{25}=1$ has the length of major axis as 10 units and length of minor axis as 4 units. That means, the graph of the original equation is elongated vertically.

d.

To determine

To find: The variation caused by the equation $\frac{{\left(x-3\right)}^2}{4}+\frac{y^2}{9}=1$ in the original equation $\frac{x^2}{4}+\frac{y^2}{9}=1$ .

Answer to Problem 156RE

Shifts the graph of the original equation towards right by 3 units.

Explanation of Solution

Given:

The original equation $\frac{x^2}{4}+\frac{y^2}{9}=1$ and variation $\frac{{\left(x-3\right)}^2}{4}+\frac{y^2}{9}=1$ .

Calculation:

The original equation $\frac{x^2}{4}+\frac{y^2}{9}=1$ represents a vertical ellipse having the length of major axis as 6 units and length of minor axis as 4 units.

The new equation $\frac{{\left(x-3\right)}^2}{4}+\frac{y^2}{9}=1$ has the length of major axis as 6 units and length of minor axis as 4 units. But here, the centre of the ellipse is $\left(3,0\right)$ . That means, it shifts the graph of the original equation towards right by 3 units.