Parabola Equation from Focus and Directrix Calculator
This parabola equation from focus and directrix calculator takes a point and a line and gives you the equation of the curve, the vertex, the axis and 4p — and it accepts a slanted directrix, which most pages will not.
Give it focus (3, 6) with directrix y = 4 and you get x² − 6x − 4y + 29 = 0, vertex (3, 5). Give it focus (0, 0) with directrix y = −x + 2 and it still answers.
The Parabola Equation from Focus and Directrix Calculator Needs No Formula
This is the part that surprises people. A parabola is the set of points that are the same distance from a fixed point as from a fixed line. So writing that sentence down as algebra is the derivation:
(x − f₁)² + (y − f₂)² = (distance to the directrix)²
Square both sides at the start — two distances are equal exactly when their squares are — and every square root disappears before you have to expand anything. What is left is ordinary multiplying out.
Why One Letter Survives and the Other Cancels
With a horizontal directrix y = d, the right-hand side is (y − d)². Both sides then carry a y², they cancel, and you are left with x squared — a parabola that opens up or down.
With a vertical directrix x = d the x² terms cancel instead and y is the squared letter, so the curve opens sideways. Nothing here is being memorised. Which letter survives is decided by the cancelling, and that tells you the orientation for free.
A Worked One, Start to Finish
Focus (3, 6), directrix y = 4:
(x − 3)² + (y − 6)² = (y − 4)²
x² − 6x + 9 + y² − 12y + 36 = y² − 8y + 16
x² − 6x − 4y + 29 = 0
The vertex is halfway between (3, 6) and the line y = 4, so it is (3, 5). The gap from focus to directrix is 2, so |4p| = 4. Three answers and the only tool used was the definition.
It opens upwards, and you can see why without any working: the focus is above the directrix, and a parabola always bends towards its focus and away from its line. The sign of p carries that — positive here, negative if the focus had been below.
The Vertex Is a Midpoint, Not a Calculation
Drop a perpendicular from the focus to the directrix and take the middle of that segment. That is the vertex, and it needs no algebra at all.
It is also the fastest check on a finished answer. Put the vertex into your equation: if it does not satisfy it, the equation is wrong, and you have found that out in ten seconds rather than at the end of the question.
A Slanted Directrix Is Allowed
Nothing in the definition says the line has to run along an axis. Take one that does not and the same expansion works exactly as before — but the answer picks up an xy term, because the parabola is now tilted.
Focus (0, 0), directrix y = −x + 2. Writing the line as x + y − 2 = 0 and squaring the distance to it:
2[(x)² + (y)²] = (x + y − 2)²
x² − 2xy + y² + 4x + 4y − 4 = 0
Vertex (1/2, 1/2), and |4p| = 2√2. That curve cannot be written as y = ax² + bx + c in any arrangement, which is exactly why calculators that only handle y = d refuse it — and why the question turns up unanswered on forums.
When the Slanted Case Still Comes Out Whole
The awkward part of a slanted directrix is the distance to it, which divides by √(A² + B²). That is usually irrational — but not always.
Take the directrix 3x − 4y = 12. Here A² + B² = 25, whose square root is a whole 5, so nothing awkward survives. With focus (3, 1) the answer is 16x² + 24xy + 9y² − 78x − 146y + 106 = 0 with |4p| = 14/5. Any Pythagorean pair in the directrix does the same.
The One Arrangement With No Parabola In It
If the focus lies on the directrix, there is no curve. The points equidistant from both are just the single straight line through the focus at right angles to it.
That is worth knowing rather than discovering halfway through: it is the only position of a point and a line that gives nothing, and every other position gives exactly one parabola.
Check It on the Curve, Not in Your Working
Take any point on your finished equation and measure both distances. They must be equal, because that equality is the definition.
This catches a wrong sign, a slip in the expansion and a mistaken vertex all at once, and it does not care how you got there. Re-reading your own algebra checks whether you did what you meant to; this checks whether what you did is right.
Using the Parabola Equation from Focus and Directrix Calculator
Type the focus in the two small boxes and the directrix in the wide one, in whatever form you have it. You get the equation cleared of fractions, the vertex, the axis, |4p|, a working that expands the definition line by line, and a drawing with the two equal distances marked — tilted, when the directrix is.
Parabola Equation from Focus and Directrix Calculator FAQ
How do you find the equation of a parabola given the focus and directrix?
Where is the vertex when you know the focus and directrix?
Can the directrix be a slanted line?
What if the focus lies on the directrix?
How do you find 4p from the focus and directrix?
Write the definition, square it, expand it — and put the vertex back in to be sure.