09 Conic Sections Free Online FlashCards

What is a conic section?

A conic section is a curve formed by intersecting a plane with a double cone, represented analytically by a second-degree equation in $x$ and $y$.

How is a parabola defined by distance?

A parabola is the set of points equidistant from a fixed point, the focus, and a fixed line, the directrix.

How is an ellipse defined by distance?

An ellipse consists of points whose distances to two fixed points, the foci, have a constant sum.

How is a hyperbola defined by distance?

A hyperbola consists of points whose distances to two fixed points have a constant absolute difference.

How do squared terms classify a nonrotated conic?

For a nonrotated equation, one squared variable indicates a parabola; same-sign squared terms with unequal coefficients indicate an ellipse; opposite signs indicate a hyperbola.

What is the standard form for a horizontal parabola?

For a horizontal parabola, the standard form is $(y-k)^2=4p(x-h)$, with vertex $(h,k)$, focus $(h+p,k)$, and directrix $x=h-p$.

What does the sign of $p$ show for a vertical parabola?

For a vertical parabola, $p>0$ means it opens upward and $p<0$ means it opens downward.

What standard form results from completing the square?

Completing the square transforms $y^2-6y-4x+1=0$ into $(y-3)^2=4(x+2)$, so the vertex is $(-2,3)$ and the parabola opens right.

How does an ellipse’s larger denominator determine orientation?

In an ellipse, the larger denominator identifies the direction of the major axis: under the $x$-term means horizontal, and under the $y$-term means vertical.

What relation connects $a$, $b$, and $c$ in an ellipse?

For an ellipse, the focal-distance relation is $c^2=a^2-b^2$, where $a$ is the semimajor axis and $b$ is the semiminor axis.

What are the key points of this ellipse?

The ellipse $\frac{(x-1)^2}{25}+\frac{(y+2)^2}{9}=1$ has center $(1,-2)$, horizontal vertices $(-4,-2)$ and $(6,-2)$, and co-vertices $(1,-5)$ and $(1,1)$.

How does the positive term orient a hyperbola?

For a hyperbola, the positive squared term gives the direction of the branches: a positive $x$-term means left and right, while a positive $y$-term means up and down.