Circles
Intersection of Circles
Grade 11
Question:
<p>The angle at which the circles \((x-1)^2 + y^2 = 10\) and \(x^2 + (y-2)^2 = 5\) intersect is:</p>
<p>(a) \(\frac{\pi}{6}\)</p>
<p>(b) \(\frac{\pi}{4}\)</p>
<p>(c) \(\frac{\pi}{3}\)</p>
<p>(d) \(\frac{\pi}{2}\)</p>
Step-by-Step Solution
Key Concept: The angle of intersection is found using the cosine formula with the radii and distance between centers.
<p>The angle of intersection of two circles is the angle between their tangent lines at a point of intersection. This equals the angle between the radii at the point of intersection. For circle 1: center \(C_1 = (1,0)\), radius \(r_1 = \sqrt{10}\). For circle 2: center \(C_2 = (0,2)\), radius \(r_2 = \sqrt{5}\). Distance between centers: \(d = \sqrt{1+4} = \sqrt{5}\). Using the formula \(\cos\theta = \frac{r_1^2 + r_2^2 - d^2}{2r_1 r_2} = \frac{10 + 5 - 5}{2\sqrt{10}\sqrt{5}} = \frac{10}{2\sqrt{50}} = 0\), we get \(\theta = \frac{\pi}{2}\).</p>
Correct Answer: D