Write the equation of a circle whose diameter has endpoints (2, 7) and (4, -3).Question 2Select one:a.(x + 3)2 + (y + 2)2 = 26b.(x + 1)2 + (y - 5)2 = 89c.(x + 1)2 + (y - 5)2 = 13d.(x - 3)2 + (y - 2)2 = 26
Question
Write the equation of a circle whose diameter has endpoints (2, 7) and (4, -3).Question 2Select one:a.(x + 3)2 + (y + 2)2 = 26b.(x + 1)2 + (y - 5)2 = 89c.(x + 1)2 + (y - 5)2 = 13d.(x - 3)2 + (y - 2)2 = 26
Solution
The equation of a circle is given by (x-h)² + (y-k)² = r² where (h,k) is the center of the circle and r is the radius.
Step 1: Find the midpoint of the diameter, which will be the center of the circle. The midpoint is given by ((x₁+x₂)/2, (y₁+y₂)/2). Substituting the given points (2,7) and (4,-3), we get ((2+4)/2, (7-3)/2) = (3,2). So, the center of the circle is (3,2).
Step 2: Find the length of the diameter using the distance formula, which is √[(x₂-x₁)² + (y₂-y₁)²]. Substituting the given points, we get √[(4-2)² + (-3-7)²] = √[4 + 100] = √104. The radius of the circle is half of the diameter, so r = √104/2 = √26.
Step 3: Substitute h, k, and r into the equation of the circle. We get (x-3)² + (y-2)² = (√26)², which simplifies to (x-3)² + (y-2)² = 26.
So, the equation of the circle is (x-3)² + (y-2)² = 26. The correct answer is option d. (x - 3)² + (y - 2)² = 26.
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