Coordinate Geometry and Conics · Strategy summary

Coordinate Geometry and Conics: strategy summary

1. Distance: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Midpoint: average the coordinates. Missing endpoint: step past the midpoint, $2M - R$.
2. Slope: $\frac{y_2 - y_1}{x_2 - x_1}$, same order on top and bottom. $Ax + By = C$ has slope $-\frac{A}{B}$.
3. Parallel: equal slopes. Perpendicular: flip and change the sign; the product is $-1$. A perpendicular bisector goes through the midpoint.
4. Circle: $(x - h)^2 + (y - k)^2 = r^2$. Signs inside are opposite to the center; the right side is $r^2$. Complete the square and add to both sides.
5. Parabola: $y = a(x - h)^2 + k$ has vertex $(h, k)$ and opens up if $a > 0$. In standard form the axis is $x = -\frac{b}{2a}$. If y is squared, it opens left or right.
6. Ellipse: axes $2a$ and $2b$, where the denominators are $a^2$ and $b^2$. Hyperbola: a minus sign between the squares; the positive term sets the direction, and the vertices are a units from the center.
7. Moves: x-axis $(x, -y)$; y-axis $(-x, y)$; $y = x$ $(y, x)$; $90^\circ$ counterclockwise $(-y, x)$; $90^\circ$ clockwise $(y, -x)$; $180^\circ$ $(-x, -y)$. Do two moves in the order given.