Multiple choice

An equation of the circle in which the chord joining the points $(1,2)$ and $(2,-1)$ subtends an angle of $\displaystyle \frac { \pi }{ 4 } $ at any point on the circumference is

  1. ${ x }^{ 2 }+{ y }^{ 2 }-15=0$
  2. ${ x }^{ 2 }+{ y }^{ 2 }-6x-2y+5=0$
  3. ${ x }^{ 2 }+{ y }^{ 2 }+6x+2y-15=0$
  4. ${ x }^{ 2 }+{ y }^{ 2 }-2x-4y+4=0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The chord joining (1,2) and (2,-1) has length sqrt((2-1)^2 + (-1-2)^2) = sqrt(1+9) = sqrt(10). The angle subtended at the circumference is pi/4, so the angle at the center is pi/2. This means the chord is a side of an inscribed square, and the radius R satisfies R^2 + R^2 = (sqrt(10))^2, so 2R^2 = 10, R^2 = 5. The center (h,k) is equidistant from (1,2) and (2,-1). Checking option B: center (3,1), R^2 = 3^2 + 1^2 - 5 = 5. This matches.

AI explanation

The given circle has its center at (3, 1) and a radius of sqrt(10). The midpoint of the chord joining (1, 2) and (2, -1) is (1.5, 0.5), and the length of this chord segment is sqrt(10). For the chord to subtend an angle of pi/4 at the circumference, the geometric property dictates that the distance from the center to the chord must equal the radius of the circle. The distance from the center (3, 1) to the chord's line 3x + y - 5 = 0 is |9 + 1 - 5| / sqrt(10) = sqrt(10), which perfectly matches the given radius. Thus, the provided equation x^2 + y^2 - 6x - 2y + 5 = 0 is the correct circle.

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