Circles
Tangent to circle
Grade 11
Question:
<p>The line \(y = mx + c\) touches the circle \((x - a)^2 + (y - b)^2 = r^2\) if</p>
<p>\(\beta = ma + c\sqrt{1 + m^2}\)</p>
<p>\(m\beta = a + c\sqrt{1 + m^2}\)</p>
<p>\(\beta + c\sqrt{1 + m^2} = ma + c\)</p>
<p>\(c = a\sqrt{1 + m^2} + ma + c\)</p>
Step-by-Step Solution
Key Concept: A line is tangent to a circle if and only if the perpendicular distance from the center to the line equals the radius. Use the point-to-line distance formula and set it equal to r.
<p><strong>Step 1:</strong> Rewrite the line in standard form: mx - y + c = 0</p><p><strong>Step 2:</strong> Apply the perpendicular distance formula from point (a, b) to the line:</p><p>Distance = |ma - b + c|/√(m² + 1)</p><p><strong>Step 3:</strong> For tangency, this distance must equal the radius r:</p><p>|ma - b + c|/√(m² + 1) = r</p><p><strong>Step 4:</strong> Square both sides to eliminate the absolute value:</p><p>(ma - b + c)² = r²(m² + 1)</p><p>∴ The line touches the circle if: <strong>(c - b + ma)² = r²(m² + 1)</strong> or equivalently <strong>|c - b + ma| = r√(m² + 1)</strong></p>
Correct Answer: A