Circles

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A circle is the locus of a point which moves in a plane such that its distance from a fixed point remains constant. The fixed point is called the center and the constant distance is the radius. Stand…

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General Form: x² + y² + 2gx + 2fy + c = 0 Step 1: Complete the square (x² + 2gx + g²) + (y² + 2fy + f²) = g² + f² - c Step 2: Standard form (x + g)² + (y + f)² = g² + f² - c Center: (-g, -f) Radius…

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Given: Center (h, k) = (2, -3), Radius r = 4 Step 1: Use standard form (x - h)² + (y - k)² = r² Step 2: Substitute values (x - 2)² + (y - (-3))² = 4² (x - 2)² + (y + 3)² = 16 Step 3: Expand to gene…

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Given: x² + y² - 6x + 8y - 11 = 0 Step 1: Compare with x² + y² + 2gx + 2fy + c = 0 2g = -6 → g = -3 2f = 8 → f = 4 c = -11 Step 2: Find center and radius Center = (-g, -f) = (3, -4) Radius = √(g² + …

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Diameter Form of Circle: (x - x₁)(x - x₂) + (y - y₁)(y - y₂) = 0 Reason: If P(x, y) is any point on the circle, then angle APB = 90° (angle in semicircle) This means AP ⊥ BP, so slope of AP × slope o…

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Parametric Form of Circle: For circle with center (h, k) and radius r: x = h + r cos θ y = k + r sin θ where θ is the parameter (0 ≤ θ < 2π) For circle with center at origin: x = r cos θ, y = r sin θ…

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Method: Use general form x² + y² + 2gx + 2fy + c = 0 Step 1: Substitute each point to get 3 equations Step 2: Solve for g, f, and c Example: Points (1, 1), (2, -1), (3, 2) Substituting: 1 + 1 + 2g …

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Use S₁₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c Position of point P: • S₁₁ < 0: P is INSIDE the circle • S₁₁ = 0: P is ON the circle • S₁₁ > 0: P is OUTSIDE the circle Example: Point (2, 1) and circle x² + y²…