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Analytic Geometry: Lines and Circles · Ages 16–17

Finding the Equation of a Tangent to a Circle

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For 11th grade students. Find the equation of the tangent line to each circle at the specified point of contact. Express final equations in standard form (Ax + By + C = 0) or slope-intercept form.

  1. Find the equation of the tangent to x^2 + y^2 = 100 at the point (6, 8).

  2. Find the equation of the tangent to x^2 + y^2 = 25 at the point (0, 5).

  3. Find the equation of the tangent to (x - 2)^2 + (y + 1)^2 = 25 at the point (5, 3).

  4. Find the equation of the tangent to (x + 3)^2 + (y - 4)^2 = 20 at the point (-1, 8).

  5. Find the equation of the tangent to x^2 + y^2 - 4x + 6y - 12 = 0 at the point (5, 1).

  6. Find the equation of the tangent to (x - 1)^2 + (y - 2)^2 = 16 at the point (1, -2).

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Finding the Equation of a Tangent to a Circle

Answer key

  1. Find the equation of the tangent to x^2 + y^2 = 100 at the point (6, 8).

    Center is (0, 0). Radius slope is 8/6 = 4/3. Tangent slope is -3/4. Equation: 3x + 4y - 50 = 0 (or y = -3/4 x + 25/2).

  2. Find the equation of the tangent to x^2 + y^2 = 25 at the point (0, 5).

    Center is (0, 0). The radius from (0, 0) to (0, 5) is vertical. Therefore, the tangent is horizontal: y = 5.

  3. Find the equation of the tangent to (x - 2)^2 + (y + 1)^2 = 25 at the point (5, 3).

    Center is (2, -1). Radius slope is (3 - (-1)) / (5 - 2) = 4/3. Tangent slope is -3/4. Using (5, 3): y - 3 = -3/4(x - 5), which simplifies to 3x + 4y - 27 = 0.

  4. Find the equation of the tangent to (x + 3)^2 + (y - 4)^2 = 20 at the point (-1, 8).

    Center is (-3, 4). Radius slope is (8 - 4) / (-1 - (-3)) = 4/2 = 2. Tangent slope is -1/2. Using (-1, 8): y - 8 = -1/2(x + 1), which simplifies to x + 2y - 15 = 0.

  5. Find the equation of the tangent to x^2 + y^2 - 4x + 6y - 12 = 0 at the point (5, 1).

    Complete the square to find center (2, -3). Radius slope is (1 - (-3)) / (5 - 2) = 4/3. Tangent slope is -3/4. Using (5, 1): y - 1 = -3/4(x - 5), which simplifies to 3x + 4y - 19 = 0.

  6. Find the equation of the tangent to (x - 1)^2 + (y - 2)^2 = 16 at the point (1, -2).

    Center is (1, 2). The radius connecting (1, 2) and (1, -2) is vertical (x = 1). The tangent is perpendicular, meaning it is horizontal: y = -2.