Formula Sheet

Circles Formulas

All key formulas grouped by subtopic. Each one has a quick reminder and common mistakes to watch for.

10 formulas · 7 subtopics

General Equation of a Circle

x^2 + y^2 + 2gx + 2fy + c = 0

💡 Center is (-g, -f) and radius is sqrt(g^2 + f^2 - c). The radius is real only when g^2 + f^2 - c > 0.

⚠ Forgetting to halve the coefficients of x and y to get g and f. If the equation has 4x, then g = 2, not 4.

Standard Form of a Circle

(x - h)^2 + (y - k)^2 = r^2

💡 Center is (h, k) and radius is r. This is the most direct form when center and radius are known.

Tangent from an External Point

y - y_1 = m(x - x_1) \text{ with } c^2 = a^2(1 + m^2)

💡 For the circle x^2 + y^2 = a^2, the tangent y = mx + c requires c^2 = a^2(1 + m^2). From an external point, there are exactly two tangents.

⚠ Forgetting that there are always two tangent lines from an external point and only finding one.

Length of Tangent from External Point

L = \sqrt{S_1} = \sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}

💡 S1 is obtained by substituting the external point (x1, y1) into the circle equation. This works only when the point is outside the circle (S1 > 0).

Chord of Contact T = 0

T \equiv xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0

💡 The chord of contact from an external point (x1, y1) to the circle is obtained by replacing x^2 with xx1, y^2 with yy1, x with (x+x1)/2, and y with (y+y1)/2.

Power of a Point

\text{Power} = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c = S_1

💡 Power is positive if the point is outside, zero if on the circle, and negative if inside. It equals (distance from center)^2 - r^2.

Radical Axis of Two Circles

S_1 - S_2 = 0

💡 The radical axis is the locus of points having equal power with respect to both circles. It is always perpendicular to the line joining the centers.

⚠ Writing S1 + S2 = 0 instead of S1 - S2 = 0. The radical axis is the difference, not the sum.

Condition for Orthogonal Circles

2g_1g_2 + 2f_1f_2 = c_1 + c_2

💡 Two circles are orthogonal when their tangents at the intersection points are perpendicular. This condition comes from the Pythagorean theorem applied to the triangle formed by the two centers and a point of intersection.

Parametric Form of a Circle

x = h + r\cos\theta, \quad y = k + r\sin\theta

💡 Any point on the circle (x-h)^2 + (y-k)^2 = r^2 can be written as (h + r cos(theta), k + r sin(theta)). Useful for finding points on the circle satisfying additional conditions.

Director Circle

#10

x^2 + y^2 = 2a^2

💡 The director circle of x^2 + y^2 = a^2 is the locus of the point from which two perpendicular tangents can be drawn to the circle. Its radius is a*sqrt(2).

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