Answer to Question #145221 in Analytic Geometry for Dhruv rawat

Question

The xy-plane intersects the sphere x^2+y^2+z^2+2x+2y-z=2 in a great circle.
True or false with full explanation

Answer 1

$x^2+y^2+z^2+2x+2y-z=2,$

$(x+1)^2-1+(y+1)^2-1+(z-\frac12)^2-\frac14=2,$

$(x+1)^2+(y+1)^2+(z-\frac12)^2=4\frac14.$

The xy-plane intersects the sphere in a great circle, when coordinates of the centre are $(x_0, y_0, 0).$

We have $(-1, -1, \frac12).$

The xy-plane does not intersect the sphere in a great circle.