What is the equation of the locus of the mid-point of the line segment obtained by cutting the line x + y = p, (where p is a real number) by the coordinate axes ?

If the point (x, y) is equidistant from the points (2a, 0) and (0, 3a) where a > 0, then which one of the following is correct ?

Consider the following for the next three (03) items that follow

The Plane 6x + ky + 3z - 12 = 0 where k ≠ 0 meets the coordinate axes at A, B and C respectively. The equation of the sphere passing through the origin and A, B, C is x² + y² + z² - 2x - 3y - 4z = 0.

What is the value of k?

Consider the following for the next three (03) items that follow

The Plane 6x + ky + 3z - 12 = 0 where k ≠ 0 meets the coordinate axes at A, B and C respectively. The equation of the sphere passing through the origin and A, B, C is x² + y² + z² - 2x - 3y - 4z = 0.

If p is the perpendicular distance from the centre of the sphere to the plane, then which one of the following is correct ?