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1. Equation
\[
(x – h)^2 + (y – k)^2 = r^2
\]
where \((h, k)\) is the center and \(r\) is the radius.
2. Worked Example
Find the equation of a circle with center at \( (3, -2) \) and radius \( 5 \).
Solution:
\[
(x – 3)^2 + (y + 2)^2 = 5^2
\]
\[
(x – 3)^2 + (y + 2)^2 = 25
\]
1. Standard Equation (horizontal major axis):
\[
\frac{(x – h)^2}{a^2} + \frac{(y – k)^2}{b^2} = 1
\]
where \(a > b\), \((h, k)\) is the center, and \(a\), \(b\) are the semi-major and semi-minor axes, respectively.
2. Worked Example:
Find the equation of an ellipse with center at \((0, 0)\), semi-major axis \(a = 4\), and semi-minor axis \(b = 2\).
Solution
\[
\frac{x^2}{4^2} + \frac{y^2}{2^2} = 1
\]
\[
\frac{x^2}{16} + \frac{y^2}{4} = 1
\]
1. Standard Equation (vertical axis):
\[
(y – k) = 4p(x – h)^2
\]
where \((h, k)\) is the vertex, and \(p\) is the distance from the vertex to the focus.
2. Worked Example
Find the equation of a parabola with vertex at \( (0, 0) \) and focus at \( (0, 3) \).
Solution:
The distance from the vertex to the focus is \(p = 3\), so:
\[
y = 4(3)x^2
\]
\[
y = 12x^2
\]
1. Standard Equation (horizontal transverse axis):
\[
\frac{(x – h)^2}{a^2} – \frac{(y – k)^2}{b^2} = 1
\]
where \((h, k)\) is the center, and \(a\) and \(b\) are distances related to the shape of the hyperbola.
2. Worked Example:
Find the equation of a hyperbola with center at \( (0, 0) \), \(a = 5\), and \(b = 3\).
Solution:
\[
\frac{x^2}{5^2} – \frac{y^2}{3^2} = 1
\]
\[
\frac{x^2}{25} – \frac{y^2}{9} = 1
\]
