Question

A system of equations is made up of an ellipse and a hyperbola.

Answer 1

The equation of the ellipse and the equation of the hyperbola can be

derived from the given information and the general form of the equations.

$Part \, A: The \ equation \ of \ the \ ellipse \ is; \displaystyle \frac{x^2}{3^2} + \frac{y^2}{4^2} = 1$

$Part \, B: The \ equation \ of \ the \ hyperbola \ is; \displaystyle \frac{x^2}{(-4)^2} - \frac{y^2}{(-7)^2} = 1$

Part C: The domain of the ellipse is; [-3, 3]

The domain of the hyperbola is; (-∞, -4]

• The domains of the ellipse and the hyperbola do not intersect, therefore, the system of equation has no solution.

Reasons:

Part A: The center of the ellipse = At the origin;

The vertical major axis = 8 units

The minor axis = 6 units

The general equation for an ellipse is presented as follows;

$\displaystyle \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$

Where;

$\displaystyle a = The \ \mathbf{ semi \ major \ axis} = \frac{8}{2} = 4$

$\displaystyle b = The \ semi \ \mathbf{ minor} \ axis = \frac{6}{2} = 3$

The equation of the ellipse is therefore;

• $\displaystyle \underline{\frac{x^2}{3^2} + \frac{y^2}{4^2} = 1}$

Part B: center of the hyperbola = The origin

The transverse axis = Horizontal

The vertex of the hyperbola = (-4, 0)

$\displaystyle The \ asymptote \ are; \ y = \mathbf{ \pm\frac{7}{4} \cdot x}$

The general equation of an hyperbola is presented as follows;

$\displaystyle \mathbf{\frac{x^2}{a^2} - \frac{y^2}{b^2}} = 1$

$\displaystyle The \ asymptote \ are; \ y = \mathbf{\pm\frac{b}{a} \cdot x}$

The vertices = (a, 0), (-a, 0)

Therefore, by comparison, we have;

a = -4

b = -7

Which gives the equation of the hyperbola as follows;

• $\displaystyle \underline{ \frac{x^2}{(-4)^2} - \frac{y^2}{(-7)^2} = 1}$

The above equation can be written as follows;

$\displaystyle \frac{x^2}{(4)^2} - \frac{y^2}{(7)^2} = 1$

Part C: Given that both equations are equal, we have;

The covertices of the ellipse are; (-3, 0) and (3, 0)

The domain of the ellipse = [-3, 3]

The domain of the hyperbola = x < The negative vertex

∴ The domain of the hyperbola = x < -4 = (-∞, -4]

• The above domain of the ellipse does not extend to the hyperbola, therefore there is no solution to the system.

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