Normal to a Circle: Equation, Examples, Problems Solved

The normal to a circle at a point is the line perpendicular to the tangent at that point — equivalently, the line through the point and the circle's centre. For the circle x² + y² + 2gx + 2fy + c = 0 at point (x₁, y₁), the normal is (y − y₁) = ((y₁ + f)/(x₁ + g))(x − x₁), which is simply the line joining (x₁, y₁) to the centre (−g, −f).

Normals matter because they carry the circle's radial direction: the normal at a point of contact is the direction along which a second circle's centre must lie for the circles to touch. In optics, the normal at the point of incidence defines the angle of reflection; in gear design, tooth profiles are built so that the common normal at contact passes through the pitch point. Every tangent problem has a matching normal problem.

Key Concepts

• Definition: the normal at (x₁, y₁) is perpendicular to the tangent there, so it passes through the circle's centre.
• Equation: the line through (x₁, y₁) and (−g, −f), i.e. (y − y₁)(x₁ + g) = (x − x₁)(y₁ + f).
• Slope relation: if the tangent has slope m, the normal has slope −1/m (for non-vertical, non-horizontal tangents).
• Use: two circles touch at a point exactly when their centres and the contact point are collinear on the common normal.

Formulas

• Normal at (x₁, y₁): (y − y₁) = ((y₁ + f)/(x₁ + g))(x − x₁), where (−g, −f) is the centre.
• Vector form: the normal direction is (x₁ + g, y₁ + f), the radius vector itself.
• Units: none special — x₁, g carry the same length units.
• Assumption: (x₁, y₁) lies on the circle; the denominator x₁ + g is non-zero (otherwise the normal is the vertical line x = x₁).

Worked Example

Find the normal to the circle x² + y² = 25 at the point (3, 4).

1. Identify centre and radius: centre (0, 0), radius 5. Check the point: 9 + 16 = 25, so it lies on the circle.
2. The normal passes through the centre (0, 0) and the point (3, 4).
3. Slope = (4 − 0)/(3 − 0) = 4/3, so the normal is y = (4/3)x, i.e. 4x − 3y = 0.
4. Cross-check with the tangent: the tangent at (3, 4) is 3x + 4y = 25 with slope −3/4. The normal's slope 4/3 satisfies (4/3)(−3/4) = −1, confirming perpendicularity.

Common Misconceptions

• "The normal needs a complicated formula." For circles it is just the line through the point and the centre — the radius extended.
• "Normal and tangent are interchangeable." They are perpendicular; confusing them flips slopes and ruins contact-point problems.
• "The normal only exists at special points." Every point of a smooth circle has exactly one tangent and one normal.

Real-World Applications

• Optics: the law of reflection is measured against the normal; circular mirrors reflect rays symmetrically about it.
• Gear design: involute gear teeth are shaped so the common normal at the contact point always passes through the pitch point, giving smooth power transfer.
• Wheel–rail contact: the contact force between a train wheel and rail acts along the common normal of the two circles.
• Computer graphics: shading a sphere needs the surface normal at each visible point, computed exactly this way.

In short: a circle's normal at any point is the radius line extended — through the point and the centre. Write it with two-point form, verify it is perpendicular to the tangent, and use it whenever circles touch, reflect, or press against each other.