The intersection of the spheres x² + y² + z² + 7x - 2y - z = 13 and x² + y² + z² - 3x + 3y + 4z = 8

Q: Wait, why do the $x^2, y^2,$ and $z^2$ terms disappear? Is that always true?

That is the beauty of the Radical Plane! Because both spheres have the same coefficients for their quadratic terms (specifically, the coefficient is $1$ for $x^2, y^2,$ and $z^2$), subtracting one equation from the other will always eliminate these terms. This is exactly why the intersection is a plane and not another curved surface.

Q: Does it matter if I do $S_1 - S_2$ or $S_2 - S_1$?

Not at all! If you calculate $S_2 - S_1$, you will get $-10x + 5y + 5z + 5 = 0$. If you multiply this entire equation by $-1$, you get $10x - 5y - 5z - 5 = 0$, which is identical to the result of $S_1 - S_2$. The plane remains the same.

Q: Why can't we just use the distance formula to find the intersection?

You could, but it would be incredibly tedious. You would have to find the centers and radii, determine the distance between centers, and then find the equation of the plane perpendicular to the line joining the centers. Using the Radical Plane property ($S_1 - S_2 = 0$) is the elegant, 'JEE-smart' way to bypass all that heavy calculation.

Question

The intersection of the spheres $x^2 + y^2 + z^2 + 7x - 2y - z = 13$ and $x^2 + y^2 + z^2 - 3x + 3y + 4z = 8$ is the same as the intersection of one of the sphere and the plane

Options:

(A) $2x - y - z = 1$

(B) $x - 2y - z = 1$

(C) $x - y - 2z = 1$

(D) $x - y - z = 1$

Sigma Percentile · Accepted Answer

### The Geometry of Intersection: A Journey into the Radical Plane

Welcome, future engineer. Today, we are not just solving a problem; we are sculpting space. We are looking at two spheres, two perfect, three-dimensional entities floating in the void, and we are asking a profound question: What happens when they collide?

When these two spheres intersect, they create a common boundary—a circle. Our goal today is to find the plane that contains this circle. This is the concept of the **Radical Plane**.

### Phase 1: Visualizing the Spheres

Let us define our two spheres, $S_1$ and $S_2$. We are given:

$$S_1: x^2 + y^2 + z^2 + 7x - 2y - z - 13 = 0$$

$$S_2: x^2 + y^2 + z^2 - 3x + 3y + 4z - 8 = 0$$

Take a moment to look at these equations. Notice the symmetry. Both equations share the exact same quadratic structure: $x^2 + y^2 + z^2$.

This is the signature of a sphere. In the world of coordinate geometry, when you see two equations of this form, you are looking at two objects that are fundamentally similar in their curvature. When they intersect, they create a perfectly flat, circular intersection that must lie on a plane.

### Phase 2: The Radical Plane

Here is the secret weapon of the JEE topper: the **Radical Plane**. The intersection of two spheres $S_1 = 0$ and $S_2 = 0$ is exactly the same as the intersection of one of the spheres with the plane defined by $S_1 - S_2 = 0$.

Why does this work? Because at every point on that intersection circle, the point satisfies both $S_1 = 0$ and $S_2 = 0$. Therefore, it must also satisfy $S_1 - S_2 = 0$.

Since $S_1 - S_2$ results in a linear equation, it represents a plane. It is the common ground where both spheres agree.

### Phase 3: The Algebraic Execution

Now, let us perform the subtraction. Do not be intimidated by the length of the expressions; we are simply subtracting the second equation from the first:

$$(x^2 + y^2 + z^2 + 7x - 2y - z - 13) - (x^2 + y^2 + z^2 - 3x + 3y + 4z - 8) = 0$$

Watch the magic happen. The $x^2$, $y^2$, and $z^2$ terms are identical in both spheres. When we subtract them, they vanish completely:

$$(x^2 - x^2) + (y^2 - y^2) + (z^2 - z^2) = 0$$

This is the moment of clarity. The curvature disappears, and we are left with a linear equation. Now, let us group the remaining terms with precision:

1. **The $x$ terms:** $7x - (-3x) = 7x + 3x = 10x$
2. **The $y$ terms:** $-2y - (3y) = -5y$
3. **The $z$ terms:** $-z - (4z) = -5z$
4. **The constants:** $-13 - (-8) = -13 + 8 = -5$

Putting it all together, we get:

$$10x - 5y - 5z - 5 = 0$$

### Phase 4: The Final Simplification

We are almost at the finish line. Look at the coefficients: $10, -5, -5, -5$. They are all multiples of $5$.

To make our equation elegant and clean, we divide the entire equation by $5$:

$$\frac{10x}{5} - \frac{5y}{5} - \frac{5z}{5} - \frac{5}{5} = 0$$

$$2x - y - z - 1 = 0$$

Finally, we rearrange this to match the format of our options by moving the constant to the right side:

$$2x - y - z = 1$$

This is our **Radical Plane**. It is the plane that slices through the intersection of the two spheres. You have successfully navigated the geometry, performed the algebraic reduction, and arrived at the solution.

Animated Solution for Mathematics - Three Dimensional Geometry: The intersection of the spheres $x^{2} + y^{2} + z^{2} + 7 x - 2 y - z = 13$ and $x^{2} + y^{2} + z^{2} - 3 x + 3 y + 4 z = 8$ is the same as the intersection of one of the sphere and the plane

Select Answer:

Visualized Solution

Visualizing Spheres $S_{1}$ and $S_{2}$

The Intersection Circle

The Radical Plane $S_{1} - S_{2} = 0$

Setting up the Equation

Canceling $x^{2}, y^{2}, z^{2}$ Terms

Subtracting $x$ Terms

Subtracting $y$ Terms

Subtracting $z$ Terms

Subtracting Constant Terms

The Simplified Equation

Final Simplification

Matching the Options

The Sigma Insight: Equation of a Plane

The Geometry of Intersection

Phase 1

Phase 2

Phase 3

Phase 4

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