Analytical Geometry
Circles, tangent to a circle, equation of a circle
Equation of a Circle and Tangent Lines
Standard form of a circle with centre (a, b) and radius r:
(x − a)² + (y − b)² = r²
For centre at origin: x² + y² = r².
A tangent to a circle is perpendicular to the radius at the point of tangency.
Example
Finding the Equation of a Circle
Centre (3, −2), radius 5:
(x − 3)² + (y + 2)² = 25
Given: x² + y² − 6x + 4y − 12 = 0 (complete the square)
(x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4
(x − 3)² + (y + 2)² = 25
Centre (3, −2), r = 5
Example
Equation of a Tangent
Circle: x² + y² = 25. Find the tangent at point (3, 4).
Radius gradient from (0,0) to (3,4): m_r = 4/3
Tangent ⊥ radius: m_t = −3/4
Tangent: y − 4 = −3/4(x − 3)
y = −3x/4 + 9/4 + 4
y = −3x/4 + 25/4
Note
Remember
Complete the square to convert general form to standard form. Tangent gradient × radius gradient = −1 (perpendicular). To determine if a point is inside/on/outside: substitute into (x−a)²+(y−b)² and compare with r².
Key Vocabulary
Standard form(x−a)² + (y−b)² = r² for a circle with centre (a,b)
TangentA line that touches a circle at exactly one point
Complete the squareA method to rewrite quadratic expressions in perfect square form
RadiusThe distance from centre to any point on the circle
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Standard form
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