If `x^2 + y^2 le1`, prove that the point `(x,y,sqrt(1-x^2-y^2))` is at a distance of 1 unit from the origin.

Show that if x^2+y^2=1 , then the point (x,y,sqrt(1-x^2-y^2)) is at a distance of 1 unit from the origin.

Find the points on the line (x+2)/3=(y+1)/2=(z-3)/2 at a distance of 5 units from the point (1,3,3).

If 1/2 (e^y - e^-y) = x, prove that : dy/dx = 1/(sqrt(1+x^2))

Find the equation of the plane containing the lines 2x-y+z-3=0, 3x+y+z=5 and at a distance of 1/sqrt6 from the point (2,1,-1) .

If y = x^x , prove that y_2 - y^-1 (y_1)^2 - yx^-1 = 0

If z= x + iy, x, y real, prove that : |x|+|y| le sqrt2 |z| .

Find the equation of the planes parallel to the plane x-2y+2z-3=0 and which is at a unit distance from the point A(1,1,1) .

If ysqrt(1-x^2)+xsqrt(1-y^2)=1 , then prove that dy/dx=- sqrt((1-y^2)/(1-x^2))

If y = sin^-1x , prove that (1-x^2)y_2 - x y_1= 0

If y=[sin^-1 x]^2 , then prove that : (1 -x^2) y_2-xy_1 - 2 = 0 .