The Circle: Quick Revision

A brief note on The Circle. Read it before a test. Try each question at the end before you open its answer.

Equations

• Standard form: $(x - h)^2 + (y - k)^2 = r^2$ has centre $(h, k)$ and radius $r$.
• General form: $x^2 + y^2 + 2gx + 2fy + c = 0$ has centre $(-g, -f)$ and radius $\sqrt{g^2 + f^2 - c}$.
• Real circle: it needs $g^2 + f^2 - c > 0$, the same coefficient on $x^2$ and $y^2$, and no $xy$ term.
• Diameter form: $(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$ for the ends of a diameter.
• Three points: put each into the general form and solve the three linear equations.
• Touching the axes: the $x$-axis when $c = g^2$, the $y$-axis when $c = f^2$.

A point, a line, a tangent

• Point: $S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$ is negative inside, zero on the circle and positive outside.
• Line: at distance $d$ from the centre it cuts, touches or misses as $d < r$, $d = r$ or $d > r$.
• Chord: its length is $2\sqrt{r^2 - d^2}$, and its midpoint is the foot of the perpendicular from the centre.
• Tangent at a point of the circle: $xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0$. The normal passes through the centre.
• Tangent with slope $m$: $y = mx \pm r\sqrt{1 + m^2}$ for the circle $x^2 + y^2 = r^2$.
• Outside point: the tangent length is $\sqrt{S_1}$, the chord of contact is $T = 0$, the pair of tangents is $SS_1 = T^2$, and the director circle of $x^2 + y^2 = r^2$ is $x^2 + y^2 = 2r^2$.

A worked example

Find the length of the tangent from $(6, 2)$ to $x^2 + y^2 - 4x + 6y - 12 = 0$, and the chord of contact.

1. The number $S_1$. $36 + 4 - 24 + 12 - 12 = 16$.
2. The tangent length. $\sqrt{16} = 4$.
3. The chord of contact. Here $g = -2$ and $f = 3$. $T = 6x + 2y - 2(x + 6) + 3(y + 2) - 12 = 4x + 5y - 18$. So the chord is $4x + 5y = 18$.

Practice with new numbers

A program to try

The program finds the circle through three points and then the tangent length from a point.

Mistakes to avoid

• Wrong sign of the centre. In $(x + 2)^2 + (y - 5)^2 = 9$ the centre is $(-2, 5)$.
• Not dividing first. Divide by the coefficient of $x^2$ before reading $g$, $f$ and $c$.
• Half a chord. The chord is $2\sqrt{r^2 - d^2}$.
• Tangent length as $CP - r$. It is $\sqrt{CP^2 - r^2}$.

Check yourself

1. Find the centre and radius of $x^2 + y^2 - 6x + 4y - 12 = 0$.

2. Find the tangent to $x^2 + y^2 = 25$ at $(3, 4)$.

3. How long is the tangent from $(5, 12)$ to $x^2 + y^2 = 25$?

4. Is the point $(1, 1)$ inside the circle $x^2 + y^2 - 4x - 6y + 9 = 0$?

5. The line $y = x + c$ touches $x^2 + y^2 = 2$. Find $c$.

Sources

• NCERT, Mathematics for Class XI, Chapter 10 (Conic Sections), the section on the circle.
• OpenStax, Precalculus 2e, Chapter 12 (Analytic Geometry).