Identifying Conic Sections
Identify the type of the conic section represented by the equation \(9x^2 - 16y^2 = 144\).
Identifying Circles
$x^{2}+y^{2}-8 x+16 y-1=0$ is the equation of a circle with center $(h, k)$ and radius $r$ for:
\[
h=
\]
and
\[
k=
\]
and
\[
r=
\]
Finding a Circle Using the Center and Another Point
Write the equation of the circle in the form $(x-h)^{2}+(y-k)^{2}=r^{2}$ $x^{2}+y^{2}+6 x+2 y+3=0$.
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Finding a Circle by the Diameter End Points
Find the equation of the circle with its diameter end points at A(2, -3) and B(-1, 5).
Finding the Parabola Equation Using the Vertex and Another Point
Write the equation of the parabola with:
Vertex $(-3,6)$ and $y$-intercept -10
Finding the Properties of the Parabola
Find the vertex, focus, and directrix of the parabola given by the equation \(y^2 = 4x\).
Finding the Vertex
Find the vertex of the parabola.
\[
h(x)=-x^{2}+4 x-4
\]
Finding the Vertex Form of the Parabola
Find the vertex form of the parabola given by the equation \(y = 2x^2 - 12x + 20\).
Finding the Vertex Form of an Ellipse
Given the equation of the ellipse \(9x^2 + 4y^2 - 36x + 8y - 4 = 0\), find the vertex form.
Finding the Vertex Form of a Circle
Given the equation of a circle as \(x^2 + y^2 - 6x + 8y + 9 = 0\), find the vertex form of the circle.
Finding the Vertex Form of a Hyperbola
Given the equation of a hyperbola as \(4x^2 - 9y^2 = 36\), find the vertex form of this hyperbola.
Finding the Standard Form of a Parabola
Convert the equation to standard form by completing the square on $x$ or $y$. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola.
\[
x^{2}+12 x-12 y+12=0
\]
Finding the Expanded Form of an Ellipse
Find the expanded form of the ellipse with a center at the origin, a horizontal axis length of 10 and a vertical axis length of 6.
Finding the Expanded Form of a Circle
A circle is defined by the equation \((x - 5)^2 + (y - 7)^2 = 81\). Find the expanded form of this circle.
Finding the Expanded Form of a Hyperbola
Find the expanded form of the hyperbola with center at origin, foci at (0, ±4), and vertices at (0, ±3).