We are given a point (α, β, γ) and a circle z = 0, x^2 + y^2 = a^2. We need to find the equation of the sphere passing through the given point and the circle.


To find the equation of the sphere, we need to find the center and radius of the sphere.


Since the sphere passes through the point (α, β, γ), the coordinates of the center of the sphere will be (α, β, γ).


To find the radius, we need to find the distance between the center of the sphere and any point on the sphere. We can choose any point on the given circle. Let's choose (a, 0, 0) as it is the easiest to work with.

The distance between the center (α, β, γ) and the point (a, 0, 0) can be calculated using the distance formula:
√((α - a)^2 + (β - 0)^2 + (γ - 0)^2)

Simplifying the above expression, we get:
√((α - a)^2 + β^2 + γ^2)

Since this distance is equal to the radius of the sphere, we can write:
radius = √((α - a)^2 + β^2 + γ^2)


The equation of a sphere with center (h, k, l) and radius r is given by:
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2

Substituting the values of the center and radius we found in Step 1 and Step 2, we get:
(x - α)^2 + (y - β)^2 + (z - γ)^2 = ((α - a)^2 + β^2 + γ^2)

Expanding and simplifying the above equation, we get:
x^2 + y^2 + z^2 - 2αx - 2βy - 2γz + (α^2 + β^2 + γ^2 - a^2) = 0

Rearranging the terms, we get:
αx + βy + γz - (x^2 + y^2 + z^2 - a^2) = 0

Multiplying through by γ, we get:
γx^2 + γy^2 + γz^2 - γa^2 = z(α^2 + β^2 + γ^2 - a^2)

Finally, we can write the equation of the sphere as:
γ(x^2 + y^2 + z^2 - a^2) = z(α^2 + β^2 + γ^2 - a^2)


The equation of the sphere passing through the point (α, β, γ) and the circle z = 0, x^2 + y^2 = a^2 is γ(x^2 + y^2 + z^2 -