C: Coordinate geometry in the (x, y) plane flashcards

Topic 4: C: Coordinate geometry in the (x, y) plane

Equation of a line through (x₁, y₁) with gradient m, and the gradient of ax + by + c = 0

y − y₁ = m(x − x₁). The gradient of ax + by + c = 0 is −a/b.

Conditions for parallel and perpendicular lines

Parallel: m₁ = m₂. Perpendicular: m₁m₂ = −1.

Midpoint and distance between (x₁, y₁) and (x₂, y₂)

Midpoint ((x₁ + x₂)/2, (y₁ + y₂)/2). Distance √((x₂ − x₁)² + (y₂ − y₁)²).

Equation of a circle with centre (a, b) and radius r

(x − a)² + (y − b)² = r².

Find the centre and radius of x² + y² − 6x + 4y − 12 = 0.

(x − 3)² − 9 + (y + 2)² − 4 − 12 = 0, so (x − 3)² + (y + 2)² = 25. Centre (3, −2), radius 5.

Three circle properties used in coordinate geometry

The angle in a semicircle is a right angle. The perpendicular from the centre to a chord bisects the chord. The radius is perpendicular to the tangent at the point of contact.

How do you find the tangent to a circle at a point P on it?

Find the gradient of the radius from the centre to P. The tangent gradient is −1/(that gradient). Use y − y₁ = m(x − x₁) through P.

Parametric form of the circle with centre (a, b) and radius r

x = a + r cos θ, y = b + r sin θ.

Convert x = t², y = 2t to a Cartesian equation.

t = y/2, so x = y²/4, that is y² = 4x.