A parabola is defined as the set of points equidistant from: — A focus and a directrix.Every point on a parabola is equidistant from the focus (a point) and the directrix (a line).
In (x − h)² = 4p(y − k), the vertex is: — (h, k).The vertex of the parabola (x − h)² = 4p(y − k) is at (h, k).
For (x − h)² = 4p(y − k), the parabola opens upward when: — p > 0.When p > 0, the focus is above the vertex and the parabola opens upward.
For x² = 8y, identify p: — p = 2.x² = 8y → 4p = 8 → p = 2.
Which conic has the defining property "equidistant from a point and a line"? — Parabola.The parabola is defined by equal distance to a focus (point) and directrix (line).
In y² = 4px, the parabola opens: — Right.y² = 4px is a horizontal parabola. When p > 0, it opens to the right.
The distance from the vertex to the focus equals: — p.The vertex is at distance |p| from both the focus and the directrix.
An ellipse is the set of all points where the sum of distances from two foci is: — Constant.Every point on an ellipse has the same total distance from the two foci, equal to 2a.
The standard form of an ellipse centered at origin is: — x²/a² + y²/b² = 1.x²/a² + y²/b² = 1 is the standard form of an ellipse (with a > b > 0).
In x²/a² + y²/b² = 1 with a > b, the major axis is along: — The x-axis.The larger denominator determines the major axis direction. If a² > b², the major axis is horizontal (along x).
The length of the major axis is: — 2a.Major axis length = 2a, where a is the semi-major axis (the larger semi-axis).
The foci relationship for an ellipse is: — c² = a² − b².For an ellipse: c² = a² − b². The foci are inside the ellipse (c < a).
(x−2)²/9 + (y+1)²/4 = 1: what is the center? — (2, −1).Center of (x−h)²/a² + (y−k)²/b² = 1 is at (h, k) = (2, −1).
Which equation represents an ellipse? — x²/9 + y²/16 = 1.x²/9 + y²/16 = 1 is an ellipse (sum = 1, both positive terms). Option A is a circle, option B is a hyperbola.
For an ellipse, which is largest? — a.a > b > 0 and c < a. The semi-major axis a is always the largest value.
For x²/a² + y²/b² = 1, the minor axis length equals: — 2b.Minor axis length = 2b. The semi-minor axis is b.
A hyperbola is defined as the set of all points where the difference of distances from two foci is: — Constant.Every point on a hyperbola has the same absolute difference of distances from the two foci, equal to 2a.
The standard form x²/a² − y²/b² = 1 represents a hyperbola that opens: — Left and right.When x² is positive (first term), the hyperbola opens left and right (horizontal).
The standard form y²/a² − x²/b² = 1 represents a hyperbola that opens: — Up and down.When y² is positive (first term), the hyperbola opens up and down (vertical).
For a hyperbola, c² = ? — a² + b².For a hyperbola: c² = a² + b². The foci are outside the vertices (c > a).
The vertices of x²/9 − y²/16 = 1 are at: — (±3, 0).Horizontal hyperbola: vertices at (±a, 0) = (±3, 0).
The key visual difference between ellipse and hyperbola equations: — Ellipse has + between terms; hyperbola has −.x²/a² + y²/b² = 1 is an ellipse (+). x²/a² − y²/b² = 1 is a hyperbola (−). The minus sign creates the separation into two branches.
Compared to x²/a² + y²/b² = 1 (ellipse), the hyperbola x²/a² − y²/b² = 1 has: — Two separate branches.A hyperbola has two separate branches (one opening left, one right for horizontal orientation).
The asymptotes of a hyperbola are: — Lines the curve approaches but never reaches.Asymptotes are lines the hyperbola branches approach but never touch as they extend to infinity.