The length of the chord cut off by $y = 2x + 1$ from the circle $x^2 + y^2 = 2$ is
- $3/8$
- $8/7$
- $6/\sqrt{5}$
- $5/6$
The circle is x^2 + y^2 = 2, so radius r = sqrt(2). The distance d from the center (0,0) to the line 2x - y + 1 = 0 is |2(0) - 0 + 1| / sqrt(2^2 + (-1)^2) = 1 / sqrt(5). The half-length of the chord is sqrt(r^2 - d^2) = sqrt(2 - 1/5) = sqrt(9/5) = 3 / sqrt(5). The total length is 2 * (3 / sqrt(5)) = 6 / sqrt(5).
To find the length of the chord, use the formula 2 multiplied by the square root of r squared minus d squared, where r is the circle's radius and d is the perpendicular distance from the center to the line. The radius is the square root of 2, and the perpendicular distance d from the origin to the line 2x minus y plus 1 equals 0 is the absolute value of 1 divided by the square root of 5. Substituting these values gives 2 multiplied by the square root of 2 minus 1 fifth, which simplifies to 6 divided by the square root of 5.