Pure: Circle

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Grade 1 · Term 3

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Revise Pure: Circle fast with concise, exam-ready key points

About A/L Combined Mathematics: Combined Mathematics consists of Pure Mathematics and Applied Mathematics across two years.

• Identify the coefficients for the first circle (S₁): Compare x² + y² - 6x - 2y + 6 = 0 with x² + y² + 2gx + 2fy + c = 0
• Identify coefficients for the first circle: From x² + y² - 8x + 2y + 1 = 0, we have 2g = -8 => g = -4, 2f = 2 => f = 1, c = 1. 2. Identify coefficients for the second circle: From x² + y² + 4x - 6y + k = 0, we have 2g' = 4 => g' = 2, 2f' = -6 => f' = -3, c' = k. 3. Apply the orthogonal intersection condition: For orthogonal intersection, 2gg' + 2ff' = c + c'. 4
• Define the general equation of the unknown circle: Let the required circle be S ≡ x² + y² + 2gx + 2fy + c = 0. 2. Use the condition that the center lies on the x-axis: If the center (-g, -f) lies on the x-axis, then its y-coordinate must be 0. So, -f = 0, which means f = 0. The equation simplifies to x² + y² + 2gx + c = 0. 3. Use the condition that the circle passes through (1, 2): Substitute x=1, y=2 into the simplified equation: 1² + 2² + 2g(1) + c = 0 => 1 + 4 + 2g + c = 0 => 2g + c + 5 = 0 (Equation 1). 4. Identify coefficients for the given circle: From x² + y² + 4x - 6y + 9 = 0, we have 2g' = 4 => g' = 2, 2f' = -6 => f' = -3, c' = 9. 5. Apply the orthogonal intersection condition: For S and the given circle, 2gg' + 2ff' = c + c'. Substitute the known values: 2g(2) + 2(0)(-3) = c + 9. This simplifies to 4g = c + 9 (Equation 2). 6. Solve the system of equations (1) and (2): We have:
• Important Points to Remember

Source: Idasara knowledge pack — english/AL_SCIENCE/combined_mathematics v13, short_notes: Lesson 30 - Finds the condition for two circle to intersect or thogonally.

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