Consider the equation of a sphere x^{2}+y^{2}+z^{2}-4x-6y-8z-16=0. Which of the following statements is/are correct? 1. z-axis is tangent to the sphere. 2. The centre of the sphere lies on the plane x+y+z-9=0. Select the correct answer using the code given below:

Explanation

The center of the sphere is (2, 3, 4) and the radius is \sqrt{2^2+3^2+4^2-(-16)} = \sqrt{4+9+16+16} = \sqrt{45}. The perpendicular distance from the center to the z-axis is \sqrt{x^2+y^2} = \sqrt{2^2+3^2} = \sqrt{13}. Since the distance \sqrt{13} is not equal to the radius \sqrt{45}, the z-axis is not tangent (Statement 1 is false). Substituting the center (2, 3, 4) into the plane equation gives 2 + 3 + 4 - 9 = 0, so the center lies on the plane (Statement 2 is true).

Related questions on 3D Geometry

• Consider the points A(2,4,6), B(-2,-4,-2), C(4,6,4) and D(8,14,12). Which of the following statements is/are correct? 1. The points ...
• A plane cuts intercepts 2, 2, 1 on the coordinate axes. What are the direction cosines of the normal to the plane?
• Consider the following statements : 1. The direction ratios of y-axis can be \langle 0, 4, 0 \rangle 2. The direction ratios of a line <st...
• If L is the line with direction ratios \lt 3,-2, 6\gt and passing through (1,-1,1), then what are the coordinates of the points on $L...
• Which one of the planes is parallel to the line \frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}?