Revision sheet

Circles

• $(x - h)^2 + (y - k)^2 = r^2$: center $(h, k)$, radius $r$.
• Expanded form: complete the square in $x$ and in $y$.

Watch out: $(x + 3)$ means $h = -3$, and the right side is $r^2$.

Parabolas

• $x^2 = 4py$: focus $(0, p)$, directrix $y = -p$.
• $y^2 = 4px$: focus $(p, 0)$, directrix $x = -p$.

Watch out: divide by 4 to get $p$.

Ellipses

• $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$: vertices at distance $a$, co-vertices at $b$.
• Foci: $c^2 = a^2 - b^2$ (larger minus smaller).
• The larger denominator gives the major axis.

Watch out: minus for ellipses, plus for hyperbolas.

Hyperbolas

• Vertices on the axis of the positive term, at distance $a$.
• Foci: $c^2 = a^2 + b^2$.
• $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$: asymptotes $y = \pm\dfrac{b}{a}x$.

Watch out: plus for hyperbolas, minus for ellipses.

Which conic?

• One variable squared: parabola.
• Both, same sign: ellipse (equal coefficients: circle).
• Both, opposite signs: hyperbola.

Watch out: look at the squared terms, not the constant.

A line and a conic

1. Substitute the line into the conic.
2. Solve the quadratic: 0, 1 or 2 real solutions.
3. Find the other coordinate from the line.

Watch out: a negative square means no intersection.

Tangent lines

Tangent ⊥ radius; tangent segments from a point are equal.

Watch out: a right triangle, not a subtraction.

Spheres and density

$V = \tfrac{4}{3}\pi r^3$; density = mass/volume.

Watch out: $r^3$ for volume, $r^2$ for area.