Decide if the equation defines an ellipse, a hyperbola, a parabola, or no conic section at all.

midtlinjeg
2020-11-24
Answered

Decide if the equation defines an ellipse, a hyperbola, a parabola, or no conic section at all.

You can still ask an expert for help

diskusje5

Answered 2020-11-25
Author has **82** answers

Standard equation of ellipse:

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

Standard equation of a parabola:

${y}^{2}=4ax$

Standard equation of a Hyperbola:

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

a)

$4{x}^{2}-9{y}^{2}=12$

Divide by coefficient of square terms : 4

${x}^{2}-\frac{9}{4}{y}^{2}=3$

Divide by coefficient of square terms : 9

$\frac{1}{9}{x}^{2}-\frac{1}{4}{y}^{2}=\frac{1}{3}$

Divide by$\frac{1}{3}$

$\frac{{x}^{2}}{3}-\frac{{y}^{2}}{\frac{4}{3}}=1$

So, this is the form of hyperbola$\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$

Thus, the equation$4{x}^{2}-9{y}^{2}=12$ defines a hyperbola.

b)

$-4x+9{y}^{2}=0$

$9{y}^{2}=4x$

$y}^{2}=4\frac{x}{9$

So, this is the form of parabola${y}^{2}=4ax$

Thus, the equation$-4x+9{y}^{2}=0$ defines a parabola.

c)

$4{y}^{2}+9{x}^{2}=12$

Divide by coefficient of square terms : 9

$x}^{2}+\frac{4}{9}{y}^{2}=\frac{4}{3$

Divide by coefficient of square terms : 4

$\frac{1}{4}{x}^{2}+\frac{1}{9}{y}^{2}=\frac{1}{3}$

Divide by$\frac{1}{3}$

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

This is the form of ellipse$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$

Thus, the equation$4{y}^{2}+9{x}^{2}=12$ defines an ellipse.

d)

$4{x}^{3}+9{y}^{3}=12$

The above equation is not an ellipse, parabola and a hyperbola.

Hence, the equation$4{x}^{3}+9{y}^{3}=12$ is not a conic section.

Standard equation of a parabola:

Standard equation of a Hyperbola:

a)

Divide by coefficient of square terms : 4

Divide by coefficient of square terms : 9

Divide by

So, this is the form of hyperbola

Thus, the equation

b)

So, this is the form of parabola

Thus, the equation

c)

Divide by coefficient of square terms : 9

Divide by coefficient of square terms : 4

Divide by

This is the form of ellipse

Thus, the equation

d)

The above equation is not an ellipse, parabola and a hyperbola.

Hence, the equation

asked 2021-08-07

a)

To simplify:

The radical expression$\sqrt{20}$

To simplify:

The radical expression

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