A line and a circle

What you will be able to do

• tell whether a point is inside, on or outside a circle;
• tell whether a line cuts, touches or misses a circle, and find the length of a chord;
• write the tangent and the normal at a point of a circle;
• write a tangent with a given slope, and the line that is a tangent for a given condition.

Is the ball in the ring?

A hoop lies flat on the floor and a ball rolls near it. Is the ball inside the hoop, on it, or outside? You only need to compare one distance with one radius. We use this idea again and again in this lesson.

For a circle $S: x^2 + y^2 + 2gx + 2fy + c = 0$ and a point $P(x_1, y_1)$, write

Then $S_1 = CP^2 - r^2$, because $CP^2 = (x_1 + g)^2 + (y_1 + f)^2$. So $P$ is outside when $S_1 > 0$, on the circle when $S_1 = 0$, and inside when $S_1 < 0$. The number $S_1$ is called the power of the point $P$. It is also the square of the length of the tangent from $P$, which you meet in the next lesson.

Take the circle $x^2 + y^2 - 4x + 6y - 12 = 0$. It has centre $(2, -3)$ and radius $5$. The point $(5, 1)$ gives $25 + 1 - 20 + 6 - 12 = 0$, so it is on the circle. The point $(6, 2)$ gives $36 + 4 - 24 + 12 - 12 = 16$, so it is outside. The origin gives $-12$, so it is inside.

A line and a circle

Let the line be $ax + by + c' = 0$ and let the circle have centre $C$ and radius $r$. Find the distance from $C$ to the line. You know the formula from the chapter on straight lines:

Then

• $d < r$: the line cuts the circle in two points, and the part inside is a chord;
• $d = r$: the line touches the circle at one point, and it is a tangent;
• $d > r$: the line misses the circle.

The foot of the perpendicular from the centre to a chord is the midpoint of the chord. The radius to one end of the chord, half the chord and the perpendicular $d$ form a right triangle. So

The intercepts on the axes

Put $y = 0$ into $x^2 + y^2 + 2gx + 2fy + c = 0$. You get $x^2 + 2gx + c = 0$, with roots $-g \pm \sqrt{g^2 - c}$. The distance between the two points on the $x$-axis is the intercept on the axis:

For $x^2 + y^2 - 10x - 6y + 9 = 0$ we have $g = -5$, $f = -3$ and $c = 9$. The $x$-intercept is $2\sqrt{25 - 9} = 8$. The $y$-intercept is $2\sqrt{9 - 9} = 0$, so the circle touches the $y$-axis.

The tangent and the normal at a point

A tangent touches the circle at one point. The normal at that point is the line perpendicular to the tangent, and it passes through the centre. The radius to the point of contact is perpendicular to the tangent. You used this in the Class 10 chapter on tangents.

For the circle $S = 0$ and a point $P(x_1, y_1)$ on it, the tangent at $P$ is

The rule is simple: in $x^2 + y^2 + 2gx + 2fy + c$ replace $x^2$ by $xx_1$, $y^2$ by $yy_1$, $2x$ by $x + x_1$ and $2y$ by $y + y_1$. For the circle $x^2 + y^2 = r^2$ the tangent at $(x_1, y_1)$ is $xx_1 + yy_1 = r^2$.

Example. The tangent to $x^2 + y^2 = 25$ at $(3, 4)$ is $3x + 4y = 25$. The normal passes through the origin and $(3, 4)$, so it is $4x - 3y = 0$.

Example. For $x^2 + y^2 - 4x + 6y - 12 = 0$ at $(5, 1)$ we have $g = -2$ and $f = 3$. The rule gives $5x + y - 2(x + 5) + 3(y + 1) - 12 = 0$, which is $3x + 4y - 19 = 0$. The centre $(2, -3)$ is at distance $\dfrac{|6 - 12 - 19|}{5} = 5$ from this line, which equals the radius. The normal from $(2, -3)$ to $(5, 1)$ has slope $\dfrac{4}{3}$, and the tangent has slope $-\dfrac{3}{4}$. Their product is $-1$.

A tangent with a given slope

Let a line $y = mx + c$ touch the circle $x^2 + y^2 = r^2$. Then $d = r$:

So the tangents with slope $m$ are $y = mx \pm r\sqrt{1 + m^2}$, and the line $y = mx + c$ is a tangent exactly when $c^2 = r^2(1 + m^2)$. For a circle with centre $(h, k)$, move the origin: $y - k = m(x - h) \pm r\sqrt{1 + m^2}$.

Example. For which $k$ does $y = x + k$ touch $x^2 + y^2 = 8$? Here $m = 1$, so $k^2 = 8 \times 2 = 16$ and $k = \pm 4$.

The chord with a given midpoint

The chord of $S = 0$ whose midpoint is $(x_1, y_1)$ has the equation $T = S_1$. To see why, note that the chord is perpendicular to the line from the centre to its midpoint. For $x^2 + y^2 = 25$ and midpoint $(1, 2)$, we have $T = x + 2y - 25$ and $S_1 = 1 + 4 - 25 = -20$. So $T = S_1$ gives $x + 2y - 25 = -20$, which is $x + 2y = 5$. Its distance from the centre is $\dfrac{5}{\sqrt{5}} = \sqrt{5}$, so the chord has length $2\sqrt{25 - 5} = 4\sqrt{5}$.

Practise with new numbers

Try it in Python

The program takes a circle and a line, finds their distance, and says whether the line cuts, touches or misses. When it cuts, it finds the two ends of the chord.

Common mistakes

• Using $T = 0$ for a point that is not on the circle. For an outside point, $T = 0$ is the chord of contact, not a tangent.
• Dropping the absolute value in the distance. A distance is never negative.
• Giving half the chord. The chord is $2\sqrt{r^2 - d^2}$.
• Mixing up the tangent and the normal. The normal passes through the centre. The tangent is perpendicular to it.

Remember these

• $S_1 > 0$ outside, $S_1 = 0$ on the circle, $S_1 < 0$ inside.
• A line at distance $d$ from the centre cuts, touches or misses as $d < r$, $d = r$ or $d > r$.
• The chord length is $2\sqrt{r^2 - d^2}$, and the intercept on the $x$-axis is $2\sqrt{g^2 - c}$.
• The tangent at $(x_1, y_1)$ is $T = 0$; the normal passes through the centre.
• $y = mx + c$ touches $x^2 + y^2 = r^2$ when $c^2 = r^2(1 + m^2)$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Is the point $(1, 5)$ inside, on or outside the circle $x^2 + y^2 = 25$?

2. The line $3x + 4y = 15$ cuts the circle $x^2 + y^2 = 25$. Find the length of the chord.

3. Find the tangent to $x^2 + y^2 = 20$ at $(2, 4)$.

4. For which values of $k$ is $y = 3x + k$ a tangent to $x^2 + y^2 = 10$?

5. The circle $x^2 + y^2 - 6x - 8y = 0$ passes through the origin. Find the normal to it at the origin.

References

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