Circle
Easy Overview
Circles are everywhere — wheels, pizzas, orbits, the moon. And they're deceptively simple: just a set of points all at the same distance from a center. But when you mix circles with algebra and geometry, you get some really neat formulas. This chapter is about describing circles with equations and finding tangents that just barely kiss them.
Standard equation of a circle
The basic form: (x − h)² + (y − k)² = r². Here (h, k) is the center and r is the radius. If the center is at the origin (0,0), it simplifies to x² + y² = r². That's the equation of a circle — literally just Pythagoras applied to every point on the circumference. Any time you see x² + y² with equal coefficients, think 'circle'.
General equation and completing the square
Sometimes you get x² + y² + 2gx + 2fy + c = 0. That's the general form of a circle. To find the center and radius, complete the square — group x terms, group y terms, and make them into perfect squares. Center ends up at (−g, −f) and radius is √(g² + f² − c). The condition for a real circle is g² + f² − c > 0, else it's imaginary or a point.
Tangent to a circle
A tangent is a line that touches the circle at exactly one point. For a circle x² + y² = r², the tangent at (x₁, y₁) is xx₁ + yy₁ = r². That's the formula. The radius to that point is perpendicular to the tangent — always. If you know the slope m, the tangent equation is y = mx ± r√(1 + m²). The ± gives you two parallel tangents, one on each side.
Condition for tangency
A line y = mx + c is tangent to x² + y² = r² if c² = r²(1 + m²). That's the condition — derived by solving the line and circle together and setting the discriminant to 0 (only one intersection). In more general cases, find the perpendicular distance from the center to the line; if it equals the radius, the line is tangent.
Key Points
- •Standard circle: (x − h)² + (y − k)² = r²
- •General form: x² + y² + 2gx + 2fy + c = 0
- •Center (−g, −f), radius = √(g² + f² − c)
- •Tangent at (x₁, y₁) on x² + y² = r²: xx₁ + yy₁ = r²
- •Line y = mx + c is tangent if c² = r²(1 + m²)
- •Radius to point of tangency is perpendicular to tangent
Practice Questions
- Find the equation of a circle with center (3, −4) and radius 5.
- Find the center and radius of x² + y² − 6x + 8y − 11 = 0.
- Find the equation of the tangent to x² + y² = 25 at the point (3, 4).
- Show that the line 3x + 4y = 25 touches the circle x² + y² = 25.
- Find the equation of a circle passing through three given points.