Parabola, Ellipse and Hyperbola

SAT Questions that focus on Parabola, Ellipse and Hyperbola require knowledge of the following topics.

Parabola

Parabola is the locus of points in the plane that are at the same distance from a line (called Directrix) and a point (called Focus).

Let's calculate, for example, the equation of a parabola in the cartesian coordinate system, with Directrix $y=0$ and Focus (a, b). If a point (x, y) is on the parabola, then its distance to the Directrix is the same as its distance to the Focus:

$y-0=\sqrt{(x-a)^2+(y-b)^2}$
$y=\sqrt{x^2-2ax+a^2+y^2-2by+b^2}$
$y^2=x^2-2ax+a^2+y^2-2by+b^2$
$y^2-y^2+2by=x^2-2ax+a^2+b^2$
$2by=x^2-2ax+a^2+b^2$
$y=(x^2-2ax+a^2+b^2)/2b$

In other words, the equation of a parabola with a Directrix parallel to the x-axis is a quadratic equation.

Ellipse

Ellipse is a set of points in the plane, such that the sum of the distances to two fixed points (called Focal Points) is constant.

Consider an elipse centered at the origin of a cartesian coordinate system (0, 0), with width  "2a" and height "2b", and Focal Points at (-c, 0) and (c, 0), as shown in the following figure:

Point (a, 0) is on the ellipse, thus the sum of its distances to the Focal Points is the constant:

$Constant=(a-c)+(a-(-c))=a-c+a+c=2a$

Point (0, b) is also on the ellipse, thus:

$\sqrt{b^2+c^2}+\sqrt{b^2+(-c)^2}=2a$
$\sqrt{b^2+c^2}+\sqrt{b^2+c^2}=2a$
$2\sqrt{b^2+c^2}=2a$
$\sqrt{b^2+c^2}=a$
$b^2+c^2=a^2$

$c^2=a^2-b^2$, for $a\geq{b}$

The equation of this standard ellipse is:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

The ratio $e=\frac{c}{a}$ is called linear eccentricity. The closer the linear eccentricity is to 1, the more elongated the ellipse is. Whe the eccentricity equals 0, the shape is a perfect circle.

Hyperbola

Hyperbola is a set of points in the plane, such that the absolute difference of the distances to two fixed points ($F_1$ and $F_2$, called Focal Points), is a constant ($2a$).

The midpoint of the line joining the two focal points is called the Center of the Hyperbola (M).

The line through the two focal points is called Major Axis.

The Major Axis contains the two vertices ($V_1$ and $V_2$), located at a distance $a$ to the Center of the Hyperbola (M).

The distance from the Focal Points ($F_1$ and $F_2$) to the Center of the Hyperbola (M) is "c", and is called Focal Distance or Linear Eccentricity. The quotient $\frac{c}{a}$ is called Eccentricity (e).

The equation of a hyperbola in the cartesian coordinate system, with focal points on the x-axis is:

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, where $b^2=c^2-a^2$.

The equation of a hyperbola in the cartesian coordinate system, with focal points on the y-axis is:

$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$, where $b^2=c^2-a^2$.

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