The length of the chord of the circle $x^{2}+y^{2}=9$ passing through $(3, 0)$ and perpendicular to the line $y+x=0$ is
- $\dfrac {3}{\sqrt{2}}$
- $3\sqrt{2}$
- $2\sqrt{3}$
-
none of these
The line perpendicular to y + x = 0 (slope -1) passing through (3, 0) is y - 0 = 1(x - 3), or y = x - 3. The distance from the center (0, 0) to this line x - y - 3 = 0 is d = |0 - 0 - 3| / sqrt(1^2 + (-1)^2) = 3 / sqrt(2). The chord length is 2 * sqrt(r^2 - d^2) = 2 * sqrt(9 - 9/2) = 2 * sqrt(9/2) = 2 * 3 / sqrt(2) = 3 * sqrt(2).
The line perpendicular to y + x = 0 has a slope of 1, and since it passes through (3, 0), its equation is y = x - 3. The distance from the center of the circle (0,0) to this line is |0 - 0 - 3| / sqrt(1^2 + (-1)^2) = 3/sqrt(2). Using the half-chord length formula sqrt(r^2 - d^2), with radius r = 3, the half-length is sqrt(9 - 9/2) = sqrt(9/2) = 3/sqrt(2). The total length of the chord is twice this value, which is 6/sqrt(2), or 3 * sqrt(2) after rationalization. Wait, the calculation for the half-length is actually sqrt(9 - 4.5) = sqrt(4.5), and 2 * sqrt(4.5) equals 3 * sqrt(2). Let me recheck; the distance from (0,0) to y - x + 3 = 0 is |-3| / sqrt(1+1) = 3/sqrt(2). Therefore, 2 * sqrt(9 - 9/2) = 2 * sqrt(9/2) = 6/sqrt(2) = 3*sqrt(2). The answer is indeed 3*sqrt(2), but the provided option B matches this, while option A is half the length.