The length of the chord of the circle $(x-3)^{2}+(y-5)^{2}=80$ cut off by the line $3x-4y-9=0$ is
- $16$
- $8$
- $\sqrt{96}$
- $2\sqrt{96}$
The circle center is (3, 5) and radius squared is 80. The distance from (3, 5) to 3x - 4y - 9 = 0 is |3(3) - 4(5) - 9| / sqrt(3^2 + 4^2) = |9 - 20 - 9| / 5 = 20 / 5 = 4. Half-chord length = sqrt(R^2 - d^2) = sqrt(80 - 16) = sqrt(64) = 8. Full chord = 16.
The circle (x-3)^2 + (y-5)^2 = 80 has its center at (3, 5) and a radius squared of 80. The perpendicular distance from the center to the line 3x - 4y - 9 = 0 is calculated using the point-to-line distance formula: the absolute value of (9 - 20 - 9) divided by the square root of (9 + 16). This simplifies to the absolute value of -20 divided by 5, resulting in a distance of 4. Using the chord length formula 2 times the square root of (r^2 - d^2), the length is 2 times the square root of (80 - 16). Evaluating this gives 2 times the square root of 64, which is 16.