Mathematics 20 flashcards ~10 min

Coordinate Geometry and Conic Sections

About this deck

Coordinate geometry connects algebra and geometry, and this deck of 20 flashcards covers everything from basic line formulas through to the full family of conic sections. You'll start with the distance and midpoint formulas and the rules for parallel and perpendicular gradients, then move into the equation of a circle before tackling parabolas, ellipses and hyperbolas in their standard forms. The deck explains what defines each conic geometrically — for example, how an ellipse is defined by the sum of distances to two foci — along with eccentricity, which tells you how "stretched" a conic is. It finishes with practical skills like finding line-circle intersections and converting a quadratic to vertex form. Suited to A-Level and AP Precalculus, this deck is especially useful for connecting the algebraic equations to the shapes they represent, which is where many students lose marks in exams.

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𝑑 = ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2 d= (x 2 −x 1 ) 2 +(y 2 −y 1 ) 2 — derived from the Pythagorean theorem.

𝑀 = ( 𝑥 1 + 𝑥 2 2 , 𝑦 1 + 𝑦 2 2 ) M=( 2 x 1 +x 2 , 2 y 1 +y 2 ) — the average of the x-coordinates and y-coordinates.

𝑚 = 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 m= x 2 −x 1 y 2 −y 1 — change in y divided by change in x between two points.

Parallel lines have equal gradients: 𝑚 1 = 𝑚 2 m 1 =m 2 .

Perpendicular lines have gradients that are negative reciprocals of each other: 𝑚 1 × 𝑚 2 = − 1 m 1 ×m 2 =−1.

𝑥 2 + 𝑦 2 = 𝑟 2 x 2 +y 2 =r 2 , where 𝑟 r is the radius.

( 𝑥 − ℎ ) 2 + ( 𝑦 − 𝑘 ) 2 = 𝑟 2 (x−h) 2 +(y−k) 2 =r 2 .

A curve formed by the intersection of a plane and a double cone: circles, ellipses, parabolas, and hyperbolas.

𝑦 = 𝑎 𝑥 2 y=ax 2 (or 𝑥 2 = 4 𝑝 𝑦 x 2 =4py using focus-directrix form).

The focus is a fixed point, and the directrix is a fixed line, such that every point on the parabola is equidistant from both.

𝑥 2 𝑎 2 + 𝑦 2 𝑏 2 = 1 a 2 x 2 + b 2 y 2 =1, where 𝑎 a and 𝑏 b are the semi-major and semi-minor axis lengths.

The set of all points where the sum of the distances to two fixed points (the foci) is constant.

A measure of how much a conic deviates from being circular: 𝑒 = 0 e=0 for a circle, 0 < 𝑒 < 1 0 1 e>1 for a hyperbola.

𝑥 2 𝑎 2 − 𝑦 2 𝑏 2 = 1 a 2 x 2 − b 2 y 2 =1.

The set of all points where the absolute difference of the distances to two fixed foci is constant.

Straight lines that the hyperbola's branches approach but never touch as they extend to infinity, given by 𝑦 = ± 𝑏 𝑎 𝑥 y=± a b x for the standard form.

Substitute the line's equation into the circle's equation and solve the resulting quadratic; the number of real solutions indicates 0, 1 (tangent), or 2 intersection points.

It is perpendicular to the radius at that point, so its gradient is the negative reciprocal of the radius's gradient at that point.

Complete the square on the quadratic to write it as 𝑦 = 𝑎 ( 𝑥 − ℎ ) 2 + 𝑘 y=a(x−h) 2 +k, where ( ℎ , 𝑘 ) (h,k) is the vertex.

Planetary orbits (Kepler's first law describes elliptical orbits with the sun at one focus), and in engineering designs like elliptical gears and whispering galleries.