Circles

The following examples and case studies demonstrate how to apply formulas for circles, including standard and general forms, tangent lines, intersections, and orthogonal circles.

Standard and General Forms

Example

Problem 1: Find the standard equation of a circle with center C(3,−2)C(3, -2) and radius r=5r = 5. Convert it to general form.

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Example

Problem 2: Given the general equation x2+y2+8x−6y+16=0x^2 + y^2 + 8x - 6y + 16 = 0, find the center and radius of the circle.

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Example

Problem 3: Find the equation of the circle that passes through the origin (0,0)(0,0) and has its center at (4,−3)(4, -3).

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Tangent to a Circle

Example

Problem 1: Find the equation of the tangent line to the circle x2+y2=25x^2 + y^2 = 25 at the point P(3,4)P(3, 4).

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Example

Problem 2: Find the equation of the tangent to the circle (x−2)2+(y+1)2=10(x - 2)^2 + (y + 1)^2 = 10 at the point (5,0)(5, 0).

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Example

Problem 3: Find the length of the tangent segment from the external point P(7,4)P(7, 4) to the circle x2+y2+4x−6y−12=0x^2 + y^2 + 4x - 6y - 12 = 0.

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Intersection of Two Circles

Example

Problem 1: Find the equation of the radical axis (the line passing through the points of intersection) of the circles C1:x2+y2−4x−6y+4=0C_1: x^2 + y^2 - 4x - 6y + 4 = 0 and C2:x2+y2−2x+4y−6=0C_2: x^2 + y^2 - 2x + 4y - 6 = 0.

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Example

Problem 2: Determine if the circles C1:(x−1)2+(y−2)2=9C_1: (x-1)^2 + (y-2)^2 = 9 and C2:(x−6)2+(y−2)2=4C_2: (x-6)^2 + (y-2)^2 = 4 intersect.

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Orthogonal Circles

Example

Case Study 1: Prove that the circles x2+y2−4x−6y+9=0x^2 + y^2 - 4x - 6y + 9 = 0 and x2+y2+6x+4y−3=0x^2 + y^2 + 6x + 4y - 3 = 0 intersect orthogonally.

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Example

Case Study 2: Verify that the circles x2+y2−2x+4y−4=0x^2 + y^2 - 2x + 4y - 4 = 0 and x2+y2+6x+8y+12=0x^2 + y^2 + 6x + 8y + 12 = 0 are orthogonal.

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