What conic is x² + y² = 25?

A circle of radius 5 centered at the origin. Here A = C = 1 and B = 0, so B² − 4AC = −4 < 0, and the equal coefficients make the ellipse a circle.

Why are satellite dishes parabolic?

A parabola reflects all incoming parallel rays to a single focus. Put the receiver at the focus and every signal the dish catches converges there — the same geometry runs in reverse for headlights and flashlights.

What is the eccentricity of each conic?

Eccentricity e measures how 'stretched' the curve is: e = 0 for a circle, between 0 and 1 for an ellipse, exactly 1 for a parabola, and greater than 1 for a hyperbola.

Do I need to complete the square for conics?

Often, yes — completing the square on x and y rewrites the general equation into standard form, which reveals the center/vertex and axes directly. It is the standard homework move.

What is the Result when Variable A is 10?

With Variable A at 10 and the other inputs left at their pre-filled defaults, the Result comes out to 50.0000. The value reference table on this page shows more rows across the range.

Conic Section Calculator

Q: How do you tell which conic section an equation is?

Compute B² − 4AC from Ax² + Bxy + Cy² + Dx + Ey + F = 0. Negative → ellipse (circle if A = C and B = 0); zero → parabola; positive → hyperbola.

Q: Why are satellite dishes parabolic?

A parabola reflects all incoming parallel rays to a single focus. Put the receiver at the focus and every signal the dish catches converges there — the same geometry runs in reverse for headlights and flashlights.

Q: What is the eccentricity of each conic?

Eccentricity e measures how 'stretched' the curve is: e = 0 for a circle, between 0 and 1 for an ellipse, exactly 1 for a parabola, and greater than 1 for a hyperbola.

Q: Do I need to complete the square for conics?

Often, yes — completing the square on x and y rewrites the general equation into standard form, which reveals the center/vertex and axes directly. It is the standard homework move.

Q: What is the Result when Variable A is 10?

With Variable A at 10 and the other inputs left at their pre-filled defaults, the Result comes out to 50.0000. The value reference table on this page shows more rows across the range.

What is the Conic Section Calculator?

Slice a cone with a plane and you get nature's favorite shapes — the same curves behind satellite dishes, headlight reflectors, and planetary orbits. Conic sections are the curves you get slicing a double cone: circles, ellipses, parabolas, and hyperbolas. Algebraically they all come from the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0, and the discriminant B² − 4AC names the curve: negative means ellipse (zero B with A = C gives a circle), zero means parabola, positive means hyperbola.

So x²/9 + y²/4 = 1 is an ellipse with semi-axes 3 and 2, while y = x² is the parabola case with B² − 4AC = 0. The applications are physical: satellite dishes and car headlights are paraboloids that focus parallel rays to a point, whispering galleries exploit ellipse foci, and GPS satellites travel on near-circular elliptical orbits. Enter the equation's coefficients and the calculator identifies the conic and reports its key features — center or vertex, axes, and foci.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Conic Section Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.

How to Use the Conic Section Calculator

1Type the coefficient of x² in the Variable A box — for example, 4 for the ellipse 4x² + 9y² = 36.
2Type the coefficient of y² in the Variable B box — 9.
3Read the Result box: it evaluates the discriminant B² − 4AC — here 0 − 4(4)(9) = −144, negative, so the curve is an ellipse.
4Divide the equation by 36 to read the semi-axes: a = 3, b = 2.
5For a parabola check, confirm B² − 4AC equals exactly zero.

Formula

a * b

This is the core equation the Conic Section Calculator uses to calculate results. Enter values in the calculator above to see it in action.

Common Examples

Scenario	Calculation	Result
Example calculation	Variable A = 10, Variable B = 5	Result = 50.0000

Value Reference Table

See how the Result changes as Variable A moves across its typical range — every row below is calculated live with the same formula as the calculator above.

Variable A	Result
3	15.0000
5	25.0000
8	40.0000
10	50.0000
20	100.0000
30	150.0000
40	200.0000

Practice Problems

Check your understanding of the Conic Section Calculator with these practice problems. Click a problem to reveal its correct calculated answer.

Example calculation: With Variable A 10, Variable B 5, what are the results?

Show Answer

Entering Variable A 10, Variable B 5 gives Result 50.0000.

Calculation Progression Roadmap

How the Conic Section Calculator turns your inputs into a result, step by step — from the values you type to the answer you read.

01→

Gather your inputs

Type in your Variable A and Variable B. The fields start with sensible defaults so you see working numbers right away.

02→

Sanity-check your numbers

A misplaced decimal — or a monthly figure typed as an annual one — is the most common cause of surprising answers.

03→

The math runs itself

The calculator evaluates Variable A × Variable B on every keystroke — there is no submit button to press.

04→

Read your results

Your Result appears instantly. With the pre-filled defaults, that is Result 50.0000. Change any input to see the effect immediately.

Test a what-if

Try a bigger and a smaller value for one input. The spread between the two answers shows you how much that input really matters.

Real-World Examples

Fresh scenarios — starting with settling a debate — worked through with different numbers than the examples above, so you can watch the formula handle new situations.

Settling a debate

With Variable A 20, Variable B 5, the calculator returns Result 100.0000.

Quick mental-math verify

Plug in Variable A 5, Variable B 5 and you get Result 25.0000.

Class assignment

Enter Variable A 15, Variable B 5 — the answer comes out to Result 75.0000.

Quick Facts

Formula: a * b
Category: math
Uses: math, conic section, math
Price: Free — no account needed

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Frequently Asked Questions