Tangent of a Circle Calculator

THE TANGENT LINE

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Type the three coefficients, then press =.

The line found, drawn and checked

The tangent line, drawn and checked — nothing rounded.

A circle, a point, and the line that just touches

Type the circle any way your book writes it — x² + y² = 25, or x² + y² − 6x + 4y − 12 = 0, or the bracket form. Then the point. A point on the circle has one tangent, a point outside has two, and a point inside has none at all.

and the point the tangent must pass through

point

On the circle gives one tangent, outside gives two, inside gives none.

Tap a box, then use the keys

A tangent touches the circle at exactly one point, and it is always at right angles to the radius there. That one fact does the whole job: for a point on the circle the tangent is (x₀−h)(x−h) + (y₀−k)(y−k) = r², which never needs a slope — so a vertical tangent like x = 5 comes out as easily as any other. From a point outside there are two, and their slopes are usually irrational; they are left exact here rather than rounded.

Examples — tap one and it solves

The working, step by step

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This tangent of a circle calculator takes the circle and a point, and gives you the line that just touches — one line if the point sits on the circle, two if it sits outside, and a plain answer if it sits inside.

Type x² + y² = 25 with the point (3, 4) and you get 3x + 4y = 25. Type the same circle with (0, 10) and you get two tangents, of slope √3 and −√3.

What the Tangent of a Circle Calculator Reads

Other pages ask for the centre and the radius separately. Your book does not give you those — it gives you x² + y² − 6x + 4y − 12 = 0, and working the centre out yourself is half the exercise. Paste the equation as it is written and this page reads it.

Where the Point Sits Decides Everything

Before any line can be found, compare the squared distance from the centre against r² — no square root needed:

equal → the point is on the circle, one tangent
bigger → the point is outside, two tangents
smaller → the point is inside, and there is no tangent at all

That last one is not a failure, it is the answer. Every line through a point inside a circle cuts it at two places, so nothing through it can touch at one.

The Radius Does the Work, Not the Slope

A tangent meets the radius at a right angle where it touches, and that single fact gives the line directly:

(x₀ − h)(x − h) + (y₀ − k)(y − k) = r²

Look at what is not in it: any letter m. Going through the slope means dividing, and some tangents have no slope to divide by.

Every Circle Has Two Vertical Tangents

At (h + r, k) and (h − r, k) — the rightmost and leftmost points of the circle — the tangent is a vertical line, x = h ± r. This is not a rare case to be waved away in a footnote; it happens on every circle there has ever been, and a method that only produces y = mx + c cannot write those two down at all. The formula above produces them without noticing anything special has happened.

From Outside: Two Lines and One Length

The centre, the external point and either touching point make a right-angled triangle, with the radius as one leg. So Pythagoras gives the tangent length straight away:

ℓ = √(d² − r²)

Both tangents from the same point are that same length, which turns up in more exam questions than the line equations do. The two slopes come from a quadratic, so a root is usually involved — √3, not 1.732. Round it early and the line stops touching the circle: it cuts it, or misses it.

Using the Tangent of a Circle Calculator

Put the circle in the top box — any form your book uses — then the point below it. The arrow keys move between boxes. You get the line or lines, the tangent length where there is one, a drawing with the touching point marked, and a check that the centre really is r away from every line shown.

Tangent of a Circle Calculator FAQ

How do you find the tangent to a circle at a point on it?

Use the fact that the tangent is at right angles to the radius. Written out, that gives x minus h times x nought minus h, plus y minus k times y nought minus k, equals r squared. No slope is needed.

How many tangents can be drawn from a point?

Two if the point is outside the circle, one if it is on the circle, and none at all if it is inside. Every line through an inside point cuts the circle at two places.

What is the length of a tangent from an external point?

The square root of d squared minus r squared, where d is the distance from the point to the centre. Both tangents from the same point have that same length.

Can a tangent to a circle be vertical?

Yes. Every circle has two vertical tangents, at its leftmost and rightmost points. They cannot be written as y equals mx plus c, which is why a slope based method gets stuck on them.

Check where the point sits first, use the radius rather than the slope, and keep the surd to the end.