UPSC NDA Math Model Paper 5

Given: The centre of the sphere is (1, 1, 2) and it passes through the point (2, 4, 6). Let the radius be R As we know that , equation of a sphere cantered at the poin $(x_0, y_0, z_0)$ with radius $R$ is given by $(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=R^2$ Here $x_0=1, y_0=1, z_0=2$ So, the equation of sphere is: $(x-1)^2+(y-1)^2+(z-2)^2=R^2$ It is given that the sphere passes through the point $(2,4,6)$ i.e The point point $(2,4,6)$ will satisfy the equation $(x-1)^2+(y-1)^2+(z-2)^2=R^2$ $\Rightarrow (2-1)^2+(4-1)^2+(6-2)^2=R^2$ $\Rightarrow R^2=26$ $\Rightarrow (x-1)^2+(y-1)^2+(z-2)^2=26$ $\Rightarrow x^2+y^2+z^2-2x-2y-4z-20=0$ Hence, the equation of the required sphere is: $x^2+y^2+z^2-2x-2y-4z-20=0$