The tangent's equation and a circle tangent to the axes

The tangent is perpendicular to the radius at the point of contact. So the equation of the tangent at a point $\displaystyle A$ of the circle is found by the perpendicularity condition:

• Work out the slope $\displaystyle m_1$ of the radius, from the centre to $\displaystyle A$.
• The tangent’s slope $\displaystyle m_2$ satisfies $\displaystyle {m_1\cdot m_2}={-1}$, that is, $\displaystyle {m_2}={-\frac{1}{m_1}}$.
• Write the equation of the line through $\displaystyle A$ with the slope $\displaystyle m_2$, as in grade 10:

The point $\displaystyle A(3,4)$ is on the circle $\displaystyle {x^{2}+y^{2}}={25}$, since $\displaystyle {3^{2}+4^{2}}={25}$.
Find the equation of the tangent to the circle at $\displaystyle A$.

• The centre is $\displaystyle O(0,0)$. The slope of the radius $\displaystyle OA$: $\displaystyle {m_1}={\frac{4-0}{3-0}}={\frac{4}{3}}$.
• The tangent’s slope: $\displaystyle {m_2}={-\frac{3}{4}}$.
• The line through $\displaystyle A(3,4)$:

When the radius is parallel to one of the axes, the tangent is parallel to the other one – and no slope is needed.

• The circle $\displaystyle {(x-2)^{2}+(y+1)^{2}}={9}$: the centre $\displaystyle M(2,-1)$, the radius $\displaystyle 3$.
• The circle’s highest point is $\displaystyle (2,2)$, and the tangent there is $\displaystyle {y}={2}$.
• Its lowest point is $\displaystyle (2,-4)$, and the tangent there is $\displaystyle {y}={-4}$.
• At the points $\displaystyle (5,-1)$ and $\displaystyle (-1,-1)$ the tangents are vertical: $\displaystyle {x}={5}$ and $\displaystyle {x}={-1}$.

A line is a tangent to a circle if and only if it has exactly one point in common with it. So the system of the circle’s equation and the tangent’s has exactly one solution.

In the example above we substitute $\displaystyle {y}={-\frac{3}{4}x+\frac{25}{4}}$ into $\displaystyle {x^{2}+y^{2}}={25}$ and, after collecting like terms, get:

• There is one solution, $\displaystyle {x}={3}$, and then $\displaystyle {y}={4}$ – the point of contact $\displaystyle A(3,4)$.
• A line that cuts the circle gives two solutions, and a line that misses it gives none.

The centre $\displaystyle (a,b)$ is $\displaystyle |b|$ from the $\displaystyle x$-axis and $\displaystyle |a|$ from the $\displaystyle y$-axis. An axis is a tangent to the circle when the centre’s distance from it equals the radius:

• the circle touches the $\displaystyle x$-axis when $\displaystyle {R}={|b|}$;
• the circle touches the $\displaystyle y$-axis when $\displaystyle {R}={|a|}$;
• the circle touches both axes when $\displaystyle {|a|}={|b|}={R}$.

• The circle $\displaystyle {(x-3)^{2}+(y+2)^{2}}={4}$: the centre $\displaystyle (3,-2)$, the radius $\displaystyle {2}={|-2|}$ – it touches the $\displaystyle x$-axis at $\displaystyle (3,0)$ and does not meet the $\displaystyle y$-axis.
• The circle $\displaystyle {(x+4)^{2}+(y-4)^{2}}={16}$: the centre $\displaystyle (-4,4)$, the radius $\displaystyle 4$ – it touches both axes, in the second quadrant.

The point $\displaystyle A(-4,3)$ is on the circle $\displaystyle {x^{2}+y^{2}}={25}$.
Find the equation of the tangent to the circle at $\displaystyle A$.

The point $\displaystyle A(4,6)$ is on the circle $\displaystyle {(x-1)^{2}+(y-2)^{2}}={25}$.
Find the equation of the tangent to the circle at $\displaystyle A$.

The circle $\displaystyle {x^{2}+y^{2}}={2}$ and the line $\displaystyle {y}={x+2}$ are given.
Solve the system, and decide: is the line a tangent to the circle?

Is the line a tangent to the circle? If so, what is the point of contact?

The circle’s centre is $\displaystyle C(-3,5)$, and the circle touches the $\displaystyle x$-axis.
Find the circle’s equation.

A circle of radius $\displaystyle 3$ touches both axes, and its centre is in the fourth quadrant.
Find the circle’s equation.

The point $\displaystyle C(5,2)$ is on the circle $\displaystyle {(x-2)^{2}+(y-1)^{2}}={10}$.
$\displaystyle CD$ is a diameter of the circle.

Find the coordinates of the point $\displaystyle D$.

Find the equation of the tangent at $\displaystyle C$ and the equation of the tangent at $\displaystyle D$.

How do the two tangents lie with respect to each other? Explain.