Question

The coordinates of the centre of a circle are \((x - 7, 2x)\). Find the value(s) of ‘x’, if the circle passes through the point \((-9, 11)\) and has radius \(5\sqrt{2}\) units.

Hint:

Squaring both sides of the distance formula immediately helps avoid working with messy square root signs.

Answer 1

Step 1: Understanding the Concept:
The distance between the centre of a circle and any point on its circumference is equal to the radius. We use the distance formula to set up an equation.
Step 2: Key Formula or Approach:
1. Distance Formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
2. Equation: \((x_{centre} - x_{point})^2 + (y_{centre} - y_{point})^2 = r^2\)
Step 3: Detailed Explanation:
1. Let Centre \(C = (x - 7, 2x)\), Point \(P = (-9, 11)\), and \(r = 5\sqrt{2}\). 2. Square the radius: \(r^2 = (5\sqrt{2})^2 = 25 \times 2 = 50\). 3. Set up the equation: \[ [(x - 7) - (-9)]^2 + [2x - 11]^2 = 50 \] \[ [x + 2]^2 + [2x - 11]^2 = 50 \] 4. Expand the squares: \[ (x^2 + 4x + 4) + (4x^2 - 44x + 121) = 50 \] 5. Combine like terms: \[ 5x^2 - 40x + 125 = 50 \] 6. Simplify into a standard quadratic equation: \[ 5x^2 - 40x + 75 = 0 \] Divide by 5: \[ x^2 - 8x + 15 = 0 \] 7. Factorize: \((x - 3)(x - 5) = 0\).
Step 4: Final Answer:
The values of \(x\) are 3 and 5.