Geometry in the coordinate plane: circles, tangents
Algebra • Graphs
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Key concepts
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Equation of a circle with centre at the origin
A circle with centre at the origin (0,0) and radius r satisfies x^2 + y^2 = r^2. The variables x and y represent coordinates of any point on the circumference. The radius r is a non-negative constant and appears as r^2 on the right-hand side, so the equation defines all points at distance r from the origin. A point (x, y) lies on the circle if and only if x^2 + y^2 equals r^2. If the equation does not simplify to a positive r^2, the expression does not represent a real circle centred at the origin. Translation of the centre produces different standard forms, but the origin-centred form remains x^2 + y^2 = r^2.
Equation of a tangent to a circle at a given point
A tangent to a circle at a point is a straight line that touches the circle at exactly one point and is perpendicular to the radius at the point of contact. For the circle x^2 + y^2 = r^2 and a point (x1, y1) on the circle, the radius from the origin to (x1, y1) has gradient y1/x1 when x1 ≠ 0. The tangent gradient equals the negative reciprocal, giving m_t = -x1/y1 when y1 ≠ 0. The tangent line can be written as y - y1 = (-x1/y1)(x - x1) when y1 ≠ 0. A robust algebraic form of the tangent avoids division by zero: x x1 + y y1 = r^2. Substitution of (x, y) yields a linear equation. That form handles special cases: when y1 = 0, the tangent becomes x = x1; when x1 = 0, the tangent becomes y = y1. The line x x1 + y y1 = r^2 follows from differentiating or from the perpendicularity (dot product) relation between radius and tangent direction.
Coordinates and sign rules in the four quadrants
The plane divides into four quadrants by the x- and y-axes. Quadrant I contains points with x > 0 and y > 0. Quadrant II contains points with x < 0 and y > 0. Quadrant III contains points with x < 0 and y < 0. Quadrant IV contains points with x > 0 and y < 0. Points on the axes have one coordinate equal to zero and may serve as special cases in gradient and tangent calculations. Distance from the origin uses the formula sqrt(x^2 + y^2) regardless of quadrant, because squaring removes sign. Midpoint and gradient formulae operate across quadrants without modification, but gradient signs reflect quadrant position: a line between points in different quadrants may have positive or negative gradient depending on the relative coordinates.
Key notes
Important points to keep in mind