<p>The greatest distance of the point \(P(10, 7)\) from the circle \(x^2 + y^2 - 4x - 2y - 20 = 0\) is</p>
Step-by-Step Solution
Key Concept: The greatest distance from an external point to a circle equals the distance from the point to the center plus the radius. First rewrite the circle equation in standard form to find center and radius.
<p><strong>Step 1:</strong> Convert circle equation to standard form.</p><p>$x^2 + y^2 - 4x - 2y - 20 = 0$</p><p>$(x^2 - 4x + 4) + (y^2 - 2y + 1) - 4 - 1 - 20 = 0$</p><p>$(x - 2)^2 + (y - 1)^2 = 25$</p><p>Center $C = (2, 1)$ and radius $r = 5$</p><p><strong>Step 2:</strong> Find distance from P(10, 7) to center C(2, 1).</p><p>$PC = \sqrt{(10-2)^2 + (7-1)^2} = \sqrt{64 + 36} = \sqrt{100} = 10$</p><p><strong>Step 3:</strong> Since P is outside the circle (PC = 10 > r = 5), the greatest distance from P to any point on the circle is:</p><p>$\text{Maximum distance} = PC + r = 10 + 5 = 15$</p><p>∴ Answer: B</p>
Correct Answer: B