Identifying Conic Sections
Identify the type of the conic section represented by the equation \(x^2 - 4y^2 - 2x + 8y - 1 = 0\).
Identifying Circles
Identify the center and radius of the circle given by the equation \((x - 3)^2 + (y + 2)^2 = 25\).
Finding a Circle Using the Center and Another Point
Given a circle with a center at point A(-1, 3) and another point on the circle B(1, 1), find the equation of the circle.
Finding a Circle by the Diameter End Points
Find the standard form of the equation of the circle with endpoints of a diameter at the points $(5,2)$ and $(-1,6)$.
Finding the Parabola Equation Using the Vertex and Another Point
Given the vertex of a parabola is at the point (2,3) and the parabola passes through the point (1,-1), find the equation of the parabola.
Finding the Properties of the Parabola
Select the equations for the directrix lines for the hyperbola defined by the equation \( \frac{x^{2}}{20}-\frac{y^{2}}{10}=1 \)
Finding the Vertex
Find the vertex of the parabola given by the equation \(y = 2x^2 + 8x + 7\)
Finding the Vertex Form of the Parabola
Given the equation of the parabola \(y = 2x^2 - 4x + 3\), find the vertex form of the parabola.
Finding the Vertex Form of an Ellipse
Given the equation of an ellipse, \(\frac{(x-5)^2}{16} + \frac{(y+3)^2}{9} = 1\), find the vertex form of the ellipse.
Finding the Vertex Form of a Circle
Find the vertex form of the circle with the equation \(x^2 + y^2 - 6x + 8y + 9 = 0\).
Finding the Vertex Form of a Hyperbola
Find the vertex form of the hyperbola given by the equation \(16x^2 - 9y^2 = 144\).
Finding the Standard Form of a Parabola
Find the standard form of the equation of the parabola satisfying the given conditions.
Focus: $(-1,6)$; Directrix: $y=2$
Finding the Expanded Form of an Ellipse
Find the expanded form of an ellipse with a center at (3, -2), a horizontal semi-axis of 5 units, and a vertical semi-axis of 3 units.
Finding the Expanded Form of a Circle
Find the expanded form of the circle equation with center at point \((h, k) = (-3, 4)\) and radius \(r = 5\)
Finding the Expanded Form of a Hyperbola
Find the expanded form of the hyperbola with the following properties: center at (4, -3), vertices (4, 1) and (4, -7), and foci at (4, 2) and (4, -8).