Correct answer: B. 33π
The area of a circle is given by the formula A = πr², where r is the radius. To find the area, we first need to determine the radius of the circle.
The radius (r), the perpendicular distance from the center to the chord (d), and half the length of the chord form a right triangle. The radius is the hypotenuse, so the relationship is r² = d² + (chord length / 2)^2.
The chord length is given as 10, so half the chord length is 10 / 2 = 5.
Next, we must find the perpendicular distance (d) from the circle's center (3, 2) to the line y = x + 3. First, we write the line's equation in the general form Ax + By + C = 0, which is x - y + 3 = 0. The distance from a point (x₀, y₀) to this line is given by the formula d = |Ax₀ + By₀ + C| / √(A² + B²).
Plugging in the values A=1, B=-1, C=3, and the center (x₀, y₀) = (3, 2):
d = |1(3) + (-1)(2) + 3| / √(1² + (-1)²)
d = |3 - 2 + 3| / √(1 + 1)
d = |4| / √(2) = 4/√(2)
To rationalize the denominator, we multiply the numerator and denominator by √(2), which gives d = (4√(2)) / 2 = 2√(2).
Now we can use the Pythagorean theorem to find r²:
r² = d² + 5²
r² = (2√(2))² + 25
r² = (4 · 2) + 25
r² = 8 + 25 = 33.
Finally, the area of the circle is A = πr² = 33π.
Choice A is incorrect because it results from misapplying the Pythagorean theorem, treating half the chord length as the hypotenuse: 5² = r² + d², which leads to r² = 25 - 8 = 17.
Choice C is incorrect because it results from an error in the distance calculation, omitting the √(2) in the denominator, which gives a distance d=4. This leads to r² = 4² + 5² = 16 + 25 = 41.
Choice D is incorrect because it results from using the full chord length (10) instead of half the length (5) in the Pythagorean theorem: r² = d² + 10² = 8 + 100 = 108.