The circles $x^2 + y^2 + 2x + 4y - 20 = 0$ and $x^2 + y^2 = 6x - 8y + 10 = 0$
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are such that the number of common tangents on them is 2.
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are not orthogonal
- are such that the length of their common tangent is $5(12 / 5)^{1/4}$
- are such that the length of their common chord is $\sqrt[5]{\dfrac{3}{2}}$
The first circle has its center at (negative 1, negative 2) and a radius of 5, while the second circle, rewritten as x squared plus y squared minus 6x plus 8y minus 10 equals 0, has its center at (3, 4) and a radius of 5. The distance between the two centers is calculated as the square root of (3 minus negative 1) squared plus (4 minus negative 2) squared, which results in a distance between centers of approximately 7.21. Because the distance between the centers is strictly less than the sum of the radii but greater than their difference, the circles intersect at exactly two points, meaning they will have exactly two common external tangents. Therefore, the number of common tangents on them is 2.