A line and a circle: their positions

A line and a circle of radius $\displaystyle R$ meet at two points, at one or at none. The position is found in two ways:

• The distance $\displaystyle d$ of the centre from the line (by the formula): $\displaystyle d<R$ – the line cuts the circle; $\displaystyle {d}={R}$ – it is a tangent; $\displaystyle d>R$ – it misses the circle.
• The system of equations: put the line into the circle’s equation and get a quadratic equation. Its discriminant positive – two points; zero – one point, a tangent; negative – no common point.

An example: the circle $\displaystyle {(x-1)^{2}+(y+2)^{2}}={10}$ and the line $\displaystyle {x-3y+3}={0}$.

• By the distance: $\displaystyle {d}={\frac{|1-3\cdot(-2)+3|}{\sqrt{1+9}}}={\frac{10}{\sqrt{10}}}={\sqrt{10}}={R}$ – the line is a tangent to the circle.
• By the system: $\displaystyle {x}={3y-3}$, and in the circle’s equation $\displaystyle {(3y-4)^{2}+(y+2)^{2}}={10}$, that is $\displaystyle {10y^{2}-20y+10}={0}$, or $\displaystyle {(y-1)^{2}}={0}$. One solution: $\displaystyle {y}={1}$, $\displaystyle {x}={0}$ – the point of contact $\displaystyle (0,1)$.

When the line cuts the circle, it makes a chord. The perpendicular from the centre to the chord bisects it, so by Pythagoras’ theorem:

$\displaystyle \ell$ – the chord’s length, $\displaystyle d$ – the centre’s distance from the line.

An example: the circle $\displaystyle {x^{2}+y^{2}}={25}$ and the line $\displaystyle {3x+4y}={15}$: $\displaystyle {d}={\frac{15}{5}}={3}$, so $\displaystyle {\frac{\ell}{2}}={\sqrt{25-9}}={4}$, and the chord is $\displaystyle 8$.

For which values of $\displaystyle n$ does the line $\displaystyle {y}={x+n}$ cut the circle $\displaystyle {x^{2}+y^{2}}={8}$?

• The line in implicit form: $\displaystyle {x-y+n}={0}$, and $\displaystyle {d}={\frac{|n|}{\sqrt{2}}}$.
• The line cuts when $\displaystyle {d<R}={\sqrt{8}}$: $\displaystyle {|n|<\sqrt{2}\cdot\sqrt{8}}={4}$, that is $\displaystyle -4<n<4$.
• When $\displaystyle {n}={4}$ or $\displaystyle {n}={-4}$ the line is a tangent, and when $\displaystyle |n|>4$ it misses.

The circle $\displaystyle {x^{2}+y^{2}-4x+2y-20}={0}$ and the line $\displaystyle {3x+4y+18}={0}$.

What are the centre and the radius?

How do they lie? And if the line cuts – how long is the chord?

Solve each system, and say how the line and the circle lie.

Mark on the board where the circle and the line meet.

The line $\displaystyle {y}={2x+n}$ touches the circle $\displaystyle {(x-1)^{2}+(y-3)^{2}}={5}$.

Write the tangents on the board.