Circles
Equation of Circle
Grade 11
Question:
<p>The given circle equation is \(x^2 + y^2 + 2x - 4y - 4 = 0\). The centre of the given circle is (–1, 2) and its radius is 3. Find the centre of the required circle.</p>
<p>(5, 2) or (–4, 2)</p>
<p>(5, –2) or (4, 2)</p>
<p>(–5, 2) or (4, –2)</p>
<p>(5, 2) or (4, –2)</p>
Step-by-Step Solution
Key Concept: Convert the general circle equation to standard form (x−h)² + (y−k)² = r² by completing the square to extract center (h, k) and radius r directly.
<p><strong>Step 1: Complete the square for x-terms</strong></p><p>x² + 2x = (x+1)² − 1</p><p><strong>Step 2: Complete the square for y-terms</strong></p><p>y² − 4y = (y−2)² − 4</p><p><strong>Step 3: Substitute back into the equation</strong></p><p>(x+1)² − 1 + (y−2)² − 4 − 4 = 0</p><p>(x+1)² + (y−2)² = 9</p><p><strong>Step 4: Extract centre and radius</strong></p><p>Comparing with (x−h)² + (y−k)² = r²:</p><p>Centre = (−1, 2) and radius = 3</p><p>∴ Answer: A</p>
Correct Answer: A