Circles
Common tangents to two circles
Grade 11
Question:
<p>For the two circles \(x^2 + y^2 = 16\) and \(x^2 + y^2 - 2y = 0\), there is/are</p>
<p>one pair of common tangents.</p>
<p>two pairs of common tangents.</p>
<p>three common tangents.</p>
<p>no common tangent.</p>
Step-by-Step Solution
Key Concept: Find the distance between centers and compare with sum/difference of radii to determine the geometric relationship (intersection type) between two circles.
<p><strong>Step 1:</strong> Identify the first circle: $x^2 + y^2 = 16$ has center $C_1 = (0, 0)$ and radius $r_1 = 4$.</p><p><strong>Step 2:</strong> Rewrite the second circle by completing the square: $x^2 + y^2 - 2y = 0 \Rightarrow x^2 + (y-1)^2 = 1$ has center $C_2 = (0, 1)$ and radius $r_2 = 1$.</p><p><strong>Step 3:</strong> Calculate the distance between centers: $d = \sqrt{(0-0)^2 + (1-0)^2} = 1$.</p><p><strong>Step 4:</strong> Apply the intersection condition:</p><ul><li>If $d = r_1 - r_2$: circles are internally tangent</li><li>Check: $1 = 4 - 1 = 3$? No.</li><li>If $|r_1 - r_2| < d < r_1 + r_2$: circles intersect at two points</li><li>Check: $|4 - 1| = 3$ and $4 + 1 = 5$, so we need $3 < 1 < 5$? No.</li><li>Since $d = 1 < |r_1 - r_2| = 3$: the smaller circle is completely inside the larger circle.</li></ul><p><strong>Step 5:</strong> When $d < |r_1 - r_2|$, one circle is entirely inside the other with no intersection points (the smaller circle lies completely within the larger one).</p><p>∴ <strong>Answer: D</strong> (No intersection points / One circle is entirely inside the other)</p>
Correct Answer: D