Find Where Graph Is Tangent To Plane Calculator

Determine the point of tangency between z = Ax² + By² and ax + by + cz = d.

Calculator

Enter the coefficients for the surface z = Ax² + By² and the plane ax + by + cz = d to find the potential point of tangency.

Comparison of ax₀ + by₀ + cz₀ and d

What is Finding Where a Graph Is Tangent to a Plane?

Finding where a graph is tangent to a plane involves identifying a point (or points) where a given surface (represented by the graph of a function, typically z = f(x,y)) touches a plane in such a way that the plane and the tangent plane to the surface at that point are identical. At the point of tangency, the normal vector to the surface is parallel to the normal vector of the plane, and the point itself lies on both the surface and the plane.

This concept is crucial in various fields of mathematics, physics, and engineering, such as optimization problems (finding extreme values subject to constraints), understanding fields and potentials, and in geometric modeling. For a surface z = f(x,y) and a plane ax + by + cz = d, we look for a point (x₀, y₀, z₀) satisfying z₀ = f(x₀, y₀), ax₀ + by₀ + cz₀ = d, and the gradient condition related to the normal vectors.

This calculator specifically addresses the case where the surface is given by z = Ax² + By² and helps you **find where graph is tangent to plane** by calculating the potential point of tangency and checking the conditions.

Common misconceptions include assuming tangency occurs whenever the surface and plane intersect, or that there's always a single point of tangency. Depending on the shapes, there could be no points, one point, or even a curve of tangency.

Find Where Graph Is Tangent to Plane Formula and Mathematical Explanation

Let the surface be defined by z = f(x,y) and the plane by ax + by + cz = d.

The normal vector to the surface z = f(x,y) (or f(x,y) – z = 0) at a point (x, y, z) is given by <∂f/∂x, ∂f/∂y, -1>. The normal vector to the plane ax + by + cz = d is <a, b, c>.

For the surface to be tangent to the plane at a point (x₀, y₀, z₀), two conditions must be met:

1. The point (x₀, y₀, z₀) must lie on both the surface and the plane:
   • z₀ = f(x₀, y₀)
   • ax₀ + by₀ + cz₀ = d
2. The normal vectors must be parallel at (x₀, y₀, z₀): <∂f/∂x(x₀,y₀), ∂f/∂y(x₀,y₀), -1> = k <a, b, c> for some scalar k. This gives -1 = kc, so k = -1/c (assuming c ≠ 0). Therefore, ∂f/∂x(x₀,y₀) = -a/c and ∂f/∂y(x₀,y₀) = -b/c.

For our specific case, f(x,y) = Ax² + By², we have ∂f/∂x = 2Ax and ∂f/∂y = 2By. So, at the point of tangency (x₀, y₀, z₀):

• 2Ax₀ = -a/c => x₀ = -a / (2Ac) (if A, c ≠ 0)
• 2By₀ = -b/c => y₀ = -b / (2Bc) (if B, c ≠ 0)
• z₀ = Ax₀² + By₀²
• And we must verify ax₀ + by₀ + cz₀ = d.

This calculator finds x₀ and y₀ using the derivative conditions, calculates z₀ from the surface equation, and then checks if the point (x₀, y₀, z₀) lies on the plane by comparing ax₀ + by₀ + cz₀ with d. Understanding how to **find where graph is tangent to plane** involves these core steps.

Variables Table

Variables used in the find where graph is tangent to plane calculation.

Variable	Meaning	Unit	Typical Range
A, B	Coefficients of x² and y² in z=Ax²+By²	None	Non-zero real numbers
a, b, c	Coefficients of x, y, z in ax+by+cz=d	None	Real numbers (c≠0 for this method)
d	Constant term in the plane equation	None	Real numbers
x₀, y₀, z₀	Coordinates of the point of tangency	None	Calculated real numbers

Practical Examples (Real-World Use Cases)

Understanding how to find where graph is tangent to plane has practical applications.

Example 1: Satellite Dish and Signal Plane

Imagine a parabolic satellite dish represented by z = 0.05x² + 0.05y² (A=0.05, B=0.05). We want to find if it can be tangent to an incoming signal plane modeled by x + y + 20z = 10 (a=1, b=1, c=20, d=10).

Using the formulas: x₀ = -1 / (2 * 0.05 * 20) = -1 / 2 = -0.5 y₀ = -1 / (2 * 0.05 * 20) = -1 / 2 = -0.5 z₀ = 0.05*(-0.5)² + 0.05*(-0.5)² = 0.05*0.25 + 0.05*0.25 = 0.0125 + 0.0125 = 0.025

Check: ax₀ + by₀ + cz₀ = 1*(-0.5) + 1*(-0.5) + 20*(0.025) = -0.5 – 0.5 + 0.5 = -0.5. The plane constant d is 10. Since -0.5 ≠ 10, the plane is not tangent to the dish at the point derived from the normal condition.

Example 2: Optimization in Material Science

Consider an energy surface z = 2x² + 3y² (A=2, B=3) and a constraint plane 4x + 6y + z = 5 (a=4, b=6, c=1, d=5).

x₀ = -4 / (2 * 2 * 1) = -4 / 4 = -1 y₀ = -6 / (2 * 3 * 1) = -6 / 6 = -1 z₀ = 2*(-1)² + 3*(-1)² = 2 + 3 = 5

Check: ax₀ + by₀ + cz₀ = 4*(-1) + 6*(-1) + 1*(5) = -4 – 6 + 5 = -5. The plane constant d is 5. Since -5 ≠ 5, tangency under these conditions does not occur at x=-1, y=-1.

If the plane was 4x + 6y + z = -5 (d=-5), then tangency would occur at (-1, -1, 5). Learning to find where graph is tangent to plane is useful here.

How to Use This Find Where Graph Is Tangent to Plane Calculator

1. Enter Surface Coefficients: Input the values for 'A' and 'B' for the surface equation z = Ax² + By². Ensure A and B are non-zero if using the standard derived formula.
2. Enter Plane Coefficients: Input 'a', 'b', 'c', and 'd' for the plane equation ax + by + cz = d. Ensure 'c' is non-zero.
3. Observe Results: The calculator automatically computes x₀, y₀, z₀ using the tangency conditions for the normals.
4. Check Tangency Condition: The 'Value of ax₀ + by₀ + cz₀' is calculated and compared to 'd'. If they are equal (or very close, allowing for rounding), then the point (x₀, y₀, z₀) is a point of tangency.
5. Primary Result: The main result will indicate if a tangency point is found at the calculated coordinates based on the check.
6. Chart: The bar chart visually compares ax₀ + by₀ + cz₀ and d. Equal heights suggest the point lies on the plane.
7. Reset: Use the reset button to clear inputs to default values.
8. Copy: Use copy results to get the inputs, point, and check value.

This tool helps you efficiently find where graph is tangent to plane for the given surface form.

Key Factors That Affect Find Where Graph Is Tangent to Plane Results

• Coefficients A and B: These determine the curvature of the parabolic surface z=Ax²+By². Changes in A and B alter the shape and thus where it might be tangent to a plane.
• Coefficients a, b, c of the Plane: These define the orientation of the plane in 3D space. The normal vector <a, b, c> dictates the slope of the plane in different directions. For tangency, this normal must be parallel to the surface's normal. c≠0 is crucial for the z=f(x,y) formulation.
• Constant d of the Plane: This shifts the plane along its normal vector without changing its orientation. It determines whether the plane, with the required orientation for parallel normals, actually intersects the surface at the point where normals are parallel.
• Non-zero A, B, c: The formulas x₀ = -a/(2Ac), y₀ = -b/(2Bc) require A, B, and c to be non-zero. If any are zero, the method or the specific surface/plane changes, and tangency might occur under different conditions or not at all as described.
• Type of Surface: This calculator is specific to z = Ax² + By². More complex surfaces f(x,y,z)=0 would involve different partial derivatives and potentially more complex systems of equations to find the tangency point(s).
• Mathematical Precision: Small rounding errors in calculations can lead to ax₀ + by₀ + cz₀ being very close but not exactly equal to d. We check for near-equality.

Frequently Asked Questions (FAQ)

What does it mean for a graph to be tangent to a plane?

It means the graph (surface) and the plane touch at a point, and at that point, the tangent plane to the surface is the same as the given plane. Their normal vectors are parallel, and the point is on both.

What if coefficient c is zero?

If c=0, the plane is vertical (ax+by=d). The normal to z=f(x,y) is <fx, fy, -1>. For this to be parallel to <a, b, 0>, we'd need -1=k*0, which is impossible. So, a surface z=f(x,y) cannot be tangent to a vertical plane in this way. You'd need an implicit surface F(x,y,z)=0 to analyze tangency with vertical planes more generally.

What if A or B is zero?

If A=0, the surface is z=By² (a parabolic cylinder). The x-derivative is 0. If a≠0, 0=-a/c requires a=0 (if c≠0), so tangency is restricted. If A=0 and a=0, 2Ax=-a/c becomes 0=0, not defining x₀ uniquely from this equation alone.

Can there be more than one point of tangency?

Yes, depending on the surface and plane. For z=Ax²+By² and a plane, there is typically at most one point derived this way, but other surfaces can have multiple points or even curves of tangency.

How do I interpret the result if ax₀ + by₀ + cz₀ is close to d but not exactly equal?

Due to floating-point arithmetic, very small differences are expected. If the difference is extremely small (e.g., less than 1e-9), it's likely a tangency point within the precision of the calculation.

What if the calculator says "Conditions for tangency not met"?

It means either A, B, or c was zero when it shouldn't be for the formulas used, or the calculated point (x₀, y₀, z₀) where normals might be parallel does not lie on the plane (ax₀ + by₀ + cz₀ ≠ d).

Can I use this for surfaces other than z = Ax² + By²?

No, this specific calculator's formulas are derived for z = Ax² + By². For a general z = f(x,y), you'd need to solve fₓ(x,y) = -a/c and fᵧ(x,y) = -b/c for x and y, which might be more complex.

Why is it important to find where graph is tangent to plane?

It's important in optimization (like Lagrange multipliers where gradients are parallel), finding closest points, and understanding geometric relationships between surfaces and planes.