AMU 2026 — Mathematics PYQ

1. General Equation of a Sphere

The standard general equation of a sphere in three-dimensional space is given by:

$$x^2+y^2+z^2+2ux+2vy+2wz+d=0$$

Where the centre of the sphere is located at the coordinates:

$$\text{Centre}=(-u,-v,-w)$$

2. Substitute the Given Points to Find Constants

We are given that the sphere passes through four specific points. Let’s substitute them one by one into the general equation to find the values of $d,u,v,$ and $w$.

• Point 1: Passing through the origin $(0,0,0)$

  $$0^2+0^2+0^2+2u(0)+2v(0)+2w(0)+d=0⟹d=0$$

Since $d=0$, our sphere equation simplifies to:

$$x^2+y^2+z^2+2ux+2vy+2wz=0$$

• Point 2: Passing through $(a,0,0)$

  $$a^2+0^2+0^2+2u(a)+2v(0)+2w(0)=0$$

  $$a^2+2ua=0$$

  $$a(a+2u)=0$$

  Since $a\ne 0$ (for a valid non-trivial intercept), we get:

  $$a+2u=0⟹u=-\frac{a}{2}$$

• Point 3: Passing through $(0,b,0)$

  $$0^2+b^2+0^2+2u(0)+2v(b)+2w(0)=0$$

  $$b^2+2vb=0⟹v=-\frac{b}{2}$$

• Point 4: Passing through $(0,0,c)$

  $$0^2+0^2+c^2+2u(0)+2v(0)+2w(c)=0$$

  $$c^2+2wc=0⟹w=-\frac{c}{2}$$

3. Determine the Coordinates of the Centre

Now that we have computed the values for $u$, $v$, and $w$, substitute them back into the standard formula for the centre $(-u,-v,-w)$:

$$\text{Centre}=\left(-\left(-\frac{a}{2}\right),-\left(-\frac{b}{2}\right),-\left(-\frac{c}{2}\right)\right)$$

$$\text{Centre}=\left(\frac{a}{2},\frac{b}{2},\frac{c}{2}\right)$$