Enhance Your Learning with Math Conic Sections Flash Cards for quick revision
Curves obtained by intersecting a cone with a plane at different angles and distances from the vertex.
A conic section formed by intersecting a cone with a plane perpendicular to the axis of the cone. It is defined as the set of all points equidistant from a fixed center point.
A conic section formed by intersecting a cone with a plane at an angle that is less than the angle of the cone. It is defined as the set of all points such that the sum of the distances from two fixed points (foci) is constant.
The longest diameter of an ellipse, passing through the center and the two foci.
The shortest diameter of an ellipse, passing through the center and perpendicular to the major axis.
A measure of how elongated an ellipse is. It is defined as the ratio of the distance between the foci to the length of the major axis.
A conic section formed by intersecting a cone with a plane parallel to one of the sides of the cone. It is defined as the set of all points that are equidistant from a fixed point (focus) and a fixed line (directrix).
The point on a parabola that is closest to the focus.
The fixed line outside a parabola that is equidistant to all points on the parabola.
The fixed point inside a parabola that is equidistant to all points on the parabola.
A conic section formed by intersecting a cone with a plane at an angle that is greater than the angle of the cone. It is defined as the set of all points such that the difference of the distances from two fixed points (foci) is constant.
The line segment passing through the center of a hyperbola and connecting two points on the hyperbola called vertices.
The line segment passing through the center of a hyperbola and perpendicular to the transverse axis.
The lines that a hyperbola approaches but never intersects as the distance from the center increases.
The equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2.
The equation of an ellipse with center (h, k), major axis length 2a, and minor axis length 2b is ((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1.
The equation of a parabola with vertex (h, k) and focus (h + p, k) is (x - h)^2 = 4p(y - k), where p is the distance between the vertex and the focus.
The equation of a hyperbola with center (h, k), transverse axis length 2a, and conjugate axis length 2b is ((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1.
The standard form of a circle equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.
The standard form of an ellipse equation is ((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1, where (h, k) is the center, a is the semi-major axis length, and b is the semi-minor axis length.
The standard form of a parabola equation is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance between the vertex and the focus.
The standard form of a hyperbola equation is ((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1, where (h, k) is the center, a is the semi-transverse axis length, and b is the semi-conjugate axis length.
The point (h, k) that represents the center of a conic section.
The two fixed points inside a conic section that determine its shape and position.
The points on a conic section that are farthest from the center.
The fixed line outside a conic section that is equidistant to all points on the conic section.
The line that divides a conic section into two congruent halves.
The line segment passing through the focus of a conic section and perpendicular to the axis of symmetry.
A coordinate system in which each point in the plane is determined by its distance from a fixed point (pole) and its angle from a fixed line (polar axis).
The polar equation of a circle with center (r, θ) and radius a is r = a.
The polar equation of an ellipse with center (r, θ), semi-major axis length a, and semi-minor axis length b is r = (a * b) / sqrt((b * cos(θ))^2 + (a * sin(θ))^2).
The polar equation of a parabola with vertex (r, θ) and focus (r + p, θ) is r = 2p / (1 + cos(θ)), where p is the distance between the vertex and the focus.
The polar equation of a hyperbola with center (r, θ), semi-transverse axis length a, and semi-conjugate axis length b is r = (a * b) / sqrt((b * cos(θ))^2 - (a * sin(θ))^2).
Moving a conic section horizontally or vertically by adding or subtracting values to the x and y coordinates of its equation.
Changing the orientation of a conic section by rotating its equation by a certain angle.
Changing the size of a conic section by multiplying the x and y coordinates of its equation by a certain factor.
The points where two or more conic sections intersect.
A line that touches a conic section at exactly one point.
A line that is perpendicular to the tangent of a conic section at the point of tangency.
A line segment that connects two points on a conic section.
A chord that passes through the center of a conic section.
A line that a hyperbola approaches but never intersects as the distance from the center increases.
Applications of conic sections in various fields such as astronomy, engineering, architecture, and physics.
The use of conic sections to describe the orbits of celestial bodies, such as planets and comets.
The use of conic sections in designing structures, such as bridges, tunnels, and satellite dishes.
The use of conic sections in designing buildings, such as domes and arches.
The use of conic sections in analyzing physical phenomena, such as projectile motion and gravitational fields.
The use of conic sections in designing optical devices, such as lenses and mirrors.
The use of conic sections in determining the position and trajectory of objects, such as ships and aircraft.
The use of conic sections in creating and manipulating images and animations on a computer screen.
The use of conic sections in analyzing and optimizing sports movements, such as throwing, kicking, and hitting.
The use of conic sections in designing musical instruments, such as stringed instruments and horns.
Examples of conic sections in natural phenomena, such as the shape of raindrops and the trajectory of a thrown stone.