Answer to Question #158548 in Analytic Geometry for Randal Rodriguez

Question

1. What are the general equations of an ellipse?

2. What is the general equation of a circle?

3. What is the general equation of a straight line?

4. What is the general equation of a parabola that opens upward with V(-1,2)?

5. What is the equation of a hyperbola with major axis parallel to the y – axis?

Answer 1

Solution 1:

General equations of the ellipse:

Case I: Major axis is y-axis (vertical axis)

$\dfrac{(x-h)^2}{b^2}+\dfrac{(y-k)^2}{a^2}=1$ , where $a>b$ and (h, k) is centre.

Case II: Major axis is x-axis (horizontal axis)

$\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$ , where $a>b$ and (h, k) is centre.

Solution 2:

General equation of the circle:

$(x-h)^2+(y-k)^2=r^2$ , where r is radius and (h, k) is centre.

Solution 3:

General equation of straight line:

$y=mx+c$ , where m is slope or gradient and c is y-intercept

Solution 4:

The parabola opens upward means parabola's axis is parallel to y-axis.

And we know general equation of parabola parallel to y-axiswith vertex (h, k):

$y-k=a(x-h)^2$

Given, (h, k) = (-1, 2)

We get, $y-2=a(x+1)^2$

Required general equation is $y-2=a(x+1)^2$

Solution 5:

Equation of hyperbola with major axis parallel to y-axis:

$\dfrac{(y-k)^2}{a^2}-\dfrac{(x-h)^2}{b^2}=1$ , where $a>b$ and (h, k) is vertex.