Circle
Easy Overview
Of all the shapes in geometry, the circle might be the most perfect. Every point on its circumference is exactly the same distance from the centre. That simplicity makes it elegant — but don't mistake simplicity for shallowness. When circles meet straight lines, fascinating things happen. A line can cut through a circle (secant), just touch it (tangent), or miss it entirely. Each case has its own mathematical description. The standard equation of a circle is (x − h)² + (y − k)² = r², where (h, k) is the centre and r is the radius. This is basically Pythagoras' theorem applied to every point on the circle. If the centre is at (0, 0), it simplifies to x² + y² = r² — which is as clean as equations get. But circles don't always come packaged so neatly. Often you'll see x² + y² + 2gx + 2fy + c = 0, which is the general form. To extract the centre and radius from this, you complete the square separately for x and y. The centre ends up at (−g, −f) and the radius is √(g² + f² − c). If g² + f² − c < 0, the circle is 'imaginary' — no real points satisfy it. Tangents are one of the most important ideas in the circle chapter. A tangent is a line that touches the circle at exactly one point. For the circle x² + y² = r², the tangent at point (x₁, y₁) is xx₁ + yy₁ = r². There's also the slope form: y = mx ± r√(1 + m²), which gives two tangents (one above, one below) for a given slope m. The condition for a line y = mx + c to be tangent to x² + y² = r² is c² = r²(1 + m²). The normal to a circle at a point is the line perpendicular to the tangent, passing through the point of contact. Since the radius is always perpendicular to the tangent, the normal is simply the line through the centre and the point of contact. Other concepts include the length of the tangent from an external point (PT = √(d² − r²)), the power of a point (d² − r²), the chord of contact, and the director circle — the locus of points from which tangents to a given circle are perpendicular. The director circle of x² + y² = r² is x² + y² = 2r². There's also the pole-polar relationship. For a given circle, every point has a corresponding line (its polar), and every line has a corresponding point (its pole). This elegant duality appears in many geometry problems. By the end, you'll be able to find the equation of a circle given any geometric description, work with tangents and normals confidently, and apply circle properties to solve problems. The circle is a gateway to the conic sections that follow.
Standard equation of a circle (centre-radius form)
(x − h)² + (y − k)² = r², where (h, k) is the centre and r is the radius. This is the most fundamental form. For a circle centred at the origin: x² + y² = r². The equation literal says 'the distance from (x, y) to (h, k) is r', which is the very definition of a circle. This form is ideal when you know the centre and radius directly.
General equation of a circle
x² + y² + 2gx + 2fy + c = 0. This is an expanded form where the centre is (−g, −f) and the radius is √(g² + f² − c). The coefficient of x² and y² must be equal (and non-zero) — that's the telltale sign that a second-degree equation might represent a circle. There must be no xy term. If the coefficients of x² and y² differ, it's an ellipse or something else.
Completing the square — extracting centre and radius
Given a general equation like x² + y² − 6x + 8y − 11 = 0, group x terms and y terms: (x² − 6x) + (y² + 8y) = 11. Complete the square for x: (x − 3)² − 9. For y: (y + 4)² − 16. So (x − 3)² + (y + 4)² = 36. Centre = (3, −4), radius = 6. This completing-the-square technique is your main tool for dealing with general-form circles.
Circle with given endpoints of a diameter
If A(x₁, y₁) and B(x₂, y₂) are endpoints of a diameter, the circle's equation is (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0. This uses the fact that the angle subtended by a diameter at any point on the circle is 90°. The centre is the midpoint: ((x₁+x₂)/2, (y₁+y₂)/2), and the radius is half the distance AB.
Circle through three non-collinear points
Three points uniquely determine a circle (unless they're collinear). The general approach: plug each point into x² + y² + 2gx + 2fy + c = 0 to get three equations in g, f, c. Solve the system to find the circle. Alternatively, find the perpendicular bisectors of two chords; their intersection is the centre. This is a classic application of solving simultaneous equations.
Position of a point relative to a circle
A point (x₁, y₁) is inside, on, or outside the circle x² + y² + 2gx + 2fy + c = 0 depending on whether x₁² + y₁² + 2gx₁ + 2fy₁ + c is negative, zero, or positive. For the standard form (x − h)² + (y − k)² = r², compare the distance d = √((x₁−h)² + (y₁−k)²) with r: d < r (inside), d = r (on), d > r (outside).
Intersection of a line and a circle
Substitute the line equation into the circle equation. If the resulting quadratic in x (or y) has: (a) discriminant > 0 → two distinct intersection points (secant), (b) discriminant = 0 → one intersection (tangent), (c) discriminant < 0 → no intersection. The discriminant approach is the algebraic equivalent of the geometric situation.
Equation of the tangent at a point on the circle
For the circle x² + y² = r², the tangent at (x₁, y₁) is xx₁ + yy₁ = r². For the general circle x² + y² + 2gx + 2fy + c = 0, the tangent at (x₁, y₁) is xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0. This formula is derived by 'splitting' the squared terms: x² becomes xx₁, y² becomes yy₁, and linear terms are averaged.
Slope form of the tangent — tangents with a given slope
For the circle x² + y² = r², the equation of a tangent with slope m is y = mx ± r√(1 + m²). The ± gives two parallel tangents — one above and one below the centre. The condition for the line y = mx + c to be a tangent is c² = r²(1 + m²). This is derived by setting the discriminant of the intersection equation to zero.
Equation of the normal to a circle
The normal to a circle at a point is the line perpendicular to the tangent at that point. Since the radius is perpendicular to the tangent, the normal passes through the centre. For x² + y² = r², the normal at (x₁, y₁) is simply y = (y₁/x₁)x (if x₁ ≠ 0). For the general circle, the normal is the line joining (x₁, y₁) to the centre (−g, −f).
Length of the tangent from an external point
From an external point P(x₁, y₁) to the circle x² + y² + 2gx + 2fy + c = 0, the length of the tangent segment PT is √(x₁² + y₁² + 2gx₁ + 2fy₁ + c). This is also the square root of the power of P with respect to the circle. Geometrically, PT² = d² − r², where d is the distance from P to the centre.
Power of a point with respect to a circle
The power of a point P(x₁, y₁) with respect to a circle is defined as x₁² + y₁² + 2gx₁ + 2fy₁ + c. It represents: (distance from P to centre)² − r². If power > 0, P is outside (and the square root gives the tangent length). If power = 0, P is on the circle. If power < 0, P is inside.
Chord of contact
From an external point P(x₁, y₁), draw two tangents to a circle. The line joining the two points of contact is called the chord of contact. For x² + y² = r², its equation is xx₁ + yy₁ = r². For the general circle, it's xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0 — the same as the tangent equation but with P as the external point.
Director circle — where tangents are perpendicular
The director circle is the locus of a point from which tangents drawn to a given circle are perpendicular to each other. For the circle x² + y² = r², the director circle is x² + y² = 2r² — a concentric circle with radius r√2. The director circle's equation is always 'the original circle radius squared multiplied by 2'.
Pole and polar
Given a circle x² + y² = r² and a point P(x₁, y₁), the polar of P is the line xx₁ + yy₁ = r². P is the pole of this line. If P is outside the circle, its polar passes through the points of contact of tangents from P. If P is on the circle, the polar is the tangent at P. If P is inside, the polar is an imaginary line outside. This pole-polar relationship is a beautiful geometric duality.
Common tangents to two circles
Two circles can have up to four common tangents. If circles are separate (distance > r₁ + r₂), there are 4 tangents (2 direct, 2 transverse). If they touch externally, 3 common tangents. If they intersect, 2 common tangents. If they touch internally, 1 common tangent. If one lies inside the other, no common tangents. The type and count depend on the relative positions.
Key Points
- •Standard circle: (x − h)² + (y − k)² = r²
- •General form: x² + y² + 2gx + 2fy + c = 0
- •Centre from general form: (−g, −f)
- •Radius from general form: √(g² + f² − c)
- •Completing the square is how you find centre and radius from general form
- •Diameter form: (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0
- •Tangent at (x₁, y₁) on x² + y² = r²: xx₁ + yy₁ = r²
- •Slope form of tangent: y = mx ± r√(1 + m²)
- •Condition for tangency: c² = r²(1 + m²)
- •Normal at (x₁, y₁) passes through centre — the radius is perpendicular to tangent
- •Length of tangent from external point PT = √(d² − r²)
- •Chord of contact has the same equation as the tangent, with P as the external point
- •Director circle of x² + y² = r² is x² + y² = 2r²
- •Power of a point: S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c
Practice Questions
- Find the equation of a circle with centre (3, −4) and passing through the origin.
- Find the centre and radius of the circle x² + y² − 6x + 8y − 11 = 0.
- Find the equation of the tangent to the circle x² + y² = 25 at the point (3, 4).
- Show that the line 3x + 4y = 25 touches the circle x² + y² = 25, and find the point of contact.
- Find the equation of the circle passing through the points (1, 2), (2, 3), and (3, 1).
- Find the length of the tangent from the point (7, 1) to the circle x² + y² = 25.
- Find the equation of the director circle of x² + y² − 4x + 6y − 3 = 0.
- Find the equation of the circle with endpoints of a diameter at (3, 2) and (−5, 6).