Skip to content
Topos Uranos
English

The circle: tangents, relative positions and power of a point

Summary

This lesson studies the circle in relation to the points, the lines and the other circles of the plane. It measures the position of a point by means of a number, its power, which is the difference between the square of its distance to the center and the square of the radius; it decides the position of a line by comparing the distance from the center to the line with the radius, and shows that this criterion agrees with that of the discriminant. It proves that the tangent is perpendicular to the radius and derives its equation, classifies the six relative positions of two circles by means of their radical axis and proves the power of a point theorem, from which follow the intersecting chords theorem, the tangent-secant theorem, the equality of the two tangent line segments drawn from an exterior point and the radical center of three circles.

Learning objectives

  1. Determine the position of a point and that of a line with respect to a circle, by means of the power and of the distance from the center to the line, and relate this criterion to the discriminant.
  2. Obtain the equation of the tangent at a point of the circle and those of the two tangents drawn from an exterior point.
  3. Classify the relative positions of two circles by comparing the distance between their centers with the sum and the difference of their radii, and find their radical axis.
  4. Apply the power of a point theorem and its consequences to problems on chords, on tangents and on loci.

The position of a point and its power

In the lesson on ellipses and circles, the circle with center C=(h,k)C = (h, k) and radius rr was defined as the set of the points whose distance to CC is rr, its equation (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 was obtained, and it was shown how to recognize it, by completing squares, in an expanded equation x2+y2+Dx+Ey+F=0x^2 + y^2 + Dx + Ey + F = 0. This lesson starts from there. A point PP whose distance to CC is dd is interior to the circle if d<rd < r, lies on it if d=rd = r and is exterior if d>rd > r. Since dd and rr are not negative, comparing them amounts to comparing their squares, by the rules for inequalities from the lesson on the real line; it follows that the position of PP is decided by the sign of a single number.

Consider a circle with center C=(h,k)C = (h, k) and radius rr, and let P=(x0,y0)P = (x_0, y_0) be a point whose distance to CC is dd. The power of PP with respect to the circle is the number p=d2−r2p = d^2 - r^2. The point is interior if p<0p < 0, lies on the circle if p=0p = 0 and is exterior if p>0p > 0.

With the distance formula, the power is written in coordinates as follows:

p=(x0−h)2+(y0−k)2−r2p = (x_0 - h)^2 + (y_0 - k)^2 - r^2

The right-hand side is the left-hand side of the equation of the circle, written with all its terms on the left, computed at (x0,y0)(x_0, y_0). If the circle is given in expanded form, with coefficient 11 on x2x^2 and on y2y^2, that left-hand side is the same polynomial, only expanded; therefore, the power is obtained by substituting the coordinates of the point into x2+y2+Dx+Ey+Fx^2 + y^2 + Dx + Ey + F, without having to find the center or the radius. For example, for the circle x2+y2−2x−4y−20=0x^2 + y^2 - 2x - 4y - 20 = 0, with center (1,2)(1, 2) and radius 55, the point (4,6)(4, 6) has power 00 and lies on it, the point (3,5)(3, 5) has power −12-12 and is interior, and the point (7,2)(7, 2) has power 1111 and is exterior. The name power is justified in the section on the power of a point, where it will be seen that this number measures, without reference to the center, the product of two distances.