Circle Geometry and Tangents

01 / UNDERSTAND & APPLY

1. A circle is a fixed-distance locus

A circle with center C=(h,k) and radius r>0 is the set of points X=(x,y) satisfying |X−C|=r. Squaring gives (x−h)²+(y−k)²=r². The distance definition supplies the geometry; the equation permits algebraic intersection and tangent calculations.

The normalized general form is x²+y²+2gx+2fy+c=0. Completing squares gives center (−g,−f) and radius squared g²+f²−c. A positive value gives a genuine circle, zero gives a single point and a negative value gives no real points.

If the x² and y² coefficients are the same nonzero number and there is no xy term, divide by that common coefficient before reading the normalized parameters. Unequal quadratic coefficients generally describe another conic rather than a circle in the given equal-scale Cartesian axes.

A point lies outside, on or inside according as |P−C|²−r² is positive, zero or negative. This signed quantity is the power of P with respect to the circle. It is algebraically useful even for interior points, where real tangent segments do not exist.

02 / UNDERSTAND & APPLY

2. Geometric data determine the unknown coefficients

A center and radius determine the standard equation immediately. Three noncollinear points determine a unique circle: substitute them into x²+y²+Dx+Ey+F=0 and solve the three linear equations for D,E,F. Noncollinearity is essential; three distinct collinear points cannot lie on a genuine finite circle.

If A and B are endpoints of a diameter, a point X lies on the circle exactly when (X−A)·(X−B)=0. This is the right-angle condition at X, with the endpoints included algebraically. The center is the midpoint and the radius is half |AB|.

A circle tangent to a specified line has radius equal to the perpendicular distance from its center to that line. If its center is also constrained to another locus, combine these conditions. Squaring may be convenient, but radii are nonnegative and a genuine circle requires positive radius.

Whenever data produce more than one possible center, retain all configurations allowed by the wording. A circle tangent to both coordinate axes can have centers in several quadrants; restricting the center to the first quadrant is an extra condition that must come from the problem.

03 / UNDERSTAND & APPLY

3. The perpendicular foot organizes every line intersection

Let a line have perpendicular distance d from the circle center. If d>r it misses the circle; if d=r it is tangent; if d<r it cuts a chord with two distinct endpoints. This follows by splitting the radius to an endpoint into perpendicular components d and the half-chord length.

If H is the perpendicular foot from C to the line and u is a unit direction along the line, the intersection points are H±√(r²−d²)u when d≤r. The chord length is 2√(r²−d²). At tangency the square root vanishes and both points merge into H.

Algebraically, substituting a line parametrization into the circle equation gives a quadratic in the parameter. A positive discriminant means two intersections, zero means tangency and negative means no real intersection. The distance and discriminant methods encode the same geometric test.

Using a parametrized line or a normal-vector equation avoids losing vertical lines. A slope equation y=mx+b cannot represent them, so a calculation that searches over finite m alone may miss a valid vertical tangent.

04 / UNDERSTAND & APPLY

4. A tangent is perpendicular to the contact radius

At a point T on a genuine circle, the tangent is the line through T perpendicular to CT. In vector form it is (T−C)·(X−T)=0. Because |T−C|²=r², this can also be written (T−C)·(X−C)=r².

For x²+y²=r² and a contact point (x₁,y₁) on the circle, the equation is xx₁+yy₁=r². For a translated circle, use the centered coordinates: (x₁−h)(x−h)+(y₁−k)(y−k)=r².

The contact point must lie on the circle. Inserting an arbitrary off-circle point into the tangent-at-a-point expression does not produce a tangent at that point. For an exterior point it instead gives the chord of contact of its two tangents, a different object discussed below.

For normalized S=x²+y²+2gx+2fy+c=0, the tangent at an on-circle point (x₁,y₁) is xx₁+yy₁+g(x+x₁)+f(y+y₁)+c=0. Derive this from the centered form rather than memorizing a symbol-replacement rule without its contact condition.

05 / UNDERSTAND & APPLY

5. Exterior points have two tangent contacts

If P lies outside a circle, the right triangle CTP gives PT²=CP²−r². The two tangent segments from P therefore have equal length. If P is on the circle there is one tangent line; if P is inside, no real tangent line through P exists.

To find tangent lines from P, take a general line through P and impose center-to-line distance r, or impose a repeated root in the line-circle intersection. Check a vertical line separately if using slope form. Once a line is found, its contact point is the perpendicular foot from C.

For each contact T, perpendicularity PT⊥CT gives (P−T)·(T−C)=0. Rearranging with |T−C|²=r² yields (P−C)·(T−C)=r². Thus both contact points lie on the chord-of-contact line (P−C)·(X−C)=r².

The chord of contact joins the two touching points; it is not one of the tangent lines. Its midpoint lies on CP, and its distance from C is r²/CP. The same algebraic line can be defined for other P as a polar, but its interpretation as a real contact chord requires P outside.

06 / UNDERSTAND & APPLY

6. Power unifies secants and tangents

Parametrize a line through P as X=P+tu with a unit direction u. Substituting into |X−C|²=r² gives t²+2u·(P−C)t+(|P−C|²−r²)=0. By the quadratic root product, t₁t₂ equals the power CP²−r².

The parameters are directed distances along u. For an exterior secant whose two intersections lie on the same ray, the ordinary positive segment lengths multiply to the positive power. For P inside, the two intersections lie in opposite directions and the directed product is negative; the product of their ordinary lengths is the negative of the power.

At tangency the two parameter roots coincide, making the product the square of the tangent length. This recovers the right-triangle formula and explains why tangent and secant products are the same invariant.

The unit-direction condition matters in the parameter derivation. If the direction vector is not unit length, the leading quadratic coefficient is |u|², and the parameter product is power divided by |u|². Parameters and physical distances are then not numerically identical.

07 / UNDERSTAND & APPLY

7. Subtract normalized circle equations to compare powers

For two circles with distinct centers, subtract their normalized equations S₁−S₂=0. The x²+y² terms cancel, leaving the radical axis: the line of points with equal powers. It is perpendicular to the line joining the centers because its normal is parallel to their center displacement.

If the circles intersect, their intersection points have power zero to both, so the radical axis is their common chord line. If they are tangent, it is the common tangent at the touching point. If they do not meet, the radical axis can still be a real line even though there is no real common chord.

Distinct concentric circles have no points of equal power because the difference of their powers is a nonzero constant. Identical circles have equal power everywhere. These cases do not produce a unique radical axis line and must be separated from the distinct-center theorem.

At a point on the radical axis that lies outside both circles, the two tangent lengths are equal because their squares are equal powers. On an interior portion, power may be negative and there are no real tangent lengths to equate. Equal power is the general statement.

08 / UNDERSTAND & APPLY

8. Radius comparisons and circle families need exceptional cases

For radii R,r>0 and center separation d>0, the circles intersect in two points when |R−r|<d<R+r. They touch externally at d=R+r and internally at d=|R−r|. If d>R+r they are separate; if d<|R−r| one lies strictly inside the other without touching.

For distinct nonconcentric circles, the number of common tangent lines is four in the separate external case, three at external tangency, two when they intersect, one at internal tangency and zero when one is strictly inside the other. Identical circles have infinitely many common tangents and are a separate case.

A family through two fixed circle intersections can be written S₁+λ(S₁−S₂)=0, where both S expressions are normalized. The squared coefficients remain equal to 1, so each finite λ gives a circle equation through the common points. For two distinct real common points these members are genuine circles.

The alternative homogeneous form αS₁+βS₂=0 includes a line when α+β=0: that member is the radical axis rather than a circle. A parameterized family should therefore be classified after simplification, especially if the quadratic coefficient can vanish.

09 / UNDERSTAND & APPLY

9. Original practice with full geometric checks

Confirm that contact points lie on the circle and that proposed lengths fit the geometry. All circles below have positive radius unless a question asks about degeneracy.

10 / UNDERSTAND & APPLY

10. Normalize, identify the geometry and verify

Begin by putting the circle in normalized or centered form. Then decide whether the problem concerns a point position, a line intersection, a tangent contact, a power relation or a comparison of two circles. Each has a direct geometric invariant: radius, perpendicular distance or power.

The common errors are reading the center with the wrong signs, treating a radius-squared value as the radius, using the tangent-at-contact formula for an off-circle point, forgetting vertical tangents in a slope search, and confusing a contact chord with either tangent line.

For secant products, distinguish directed parameters from ordinary lengths and check whether the point is inside or outside. For radical axes, normalize both equations and separate concentric cases. For families, check whether the quadratic terms cancel.

Finish by checking existence as well as algebra. A chord cannot exceed the diameter, a tangent contact must lie on the circle, and the radius to that contact must be perpendicular to the tangent. These geometric checks can reveal inconsistent data even when a product equation has a numerical solution.