Question

The angle between the circles \(x^2+y^2−4x−6y−3=0\), \(x^2+y^2+8x−4y+11=0\)

• A. \(\frac{\pi}{2}\)
• B. \(\frac{\pi}{4}\)
• C. \(\frac{\pi}{3}\)
• D. \(\frac{\pi}{6}\)

Answer 1

The Correct Option is C

The equation \(x^2+y^2−4x−6y−3=0\) corresponds to a circle with radius \(r_1\)​, which can be calculated as \(\sqrt{2^2+3^2+3^2​}=\sqrt{16}=4\). The center of this circle is (2,3).

The equation \(x^2+y^2+8x−4y+11=0\) represents a circle with radius \(r_2​\), determined by \(\sqrt{4^2+2^2−11^2}​=\sqrt{3}\)​. Its center is (−4,2).

To find the angle \(\theta\) between these two circles, we can use the cosine formula:
\(\text{cos}(180−\theta)=\frac{2r_1​r_2​}{r_1^2​+r_2^2​−(c_1​c_2​)^2}​\)

Putting in the values:
\(\text{cos}(180−\theta)=\frac{2⋅4⋅3​}{4^2+3^2−(2⋅3)(−4⋅2)}\)
\(\text{cos}(180−\theta)=\frac{24}{25+37}=\frac{24}{62}​=\frac{12}{31}​\)

Now, to find \(\theta\):
\(\text{cos}(\theta)=\frac{1}{2}\)​
\(\theta=\frac{\pi}{3}\)​

Hence, the angle between these two circles is \(\frac{\pi}{3}\) or 60 degrees.