The tangent to a circle at a point on it

Analytic geometry, grade 12 (5 units) Lesson 7 of 21Learning

The tangent to a circle at a point on it

The tangent and the radius

In geometry we learnt: the tangent to a circle is perpendicular to the radius at the point of contact. So the tangent at a point $\displaystyle P$ on the circle is found like this:

the slope of the radius $\displaystyle MP$ ;
the tangent’s slope – the product of the slopes is $\displaystyle -1$ ;
the equation of the line through $\displaystyle P$ with this slope.

An example: the circle $\displaystyle {(x-2)^{2}+(y+1)^{2}}={25}$ and the point $\displaystyle P(5,3)$ on it ( $\displaystyle {3^{2}+4^{2}}={25}$ ).

$\displaystyle {m_{MP}}={\frac{3-(-1)}{5-2}}={\frac{4}{3}}$ , so the tangent’s slope is $\displaystyle -\frac{3}{4}$ .
$\displaystyle {y-3}={-\frac{3}{4}(x-5)}$ , in implicit form $\displaystyle {3x+4y}={27}$ .

\displaystyle (x-2)^{2}+(y+1)^{2}=25

\displaystyle 3x+4y=27

\displaystyle y=\frac{4}{3}x-\frac{11}{3}

\displaystyle M=(2,-1)

\displaystyle P=(5,3)

The tangent formula

The same three steps, in letters, give a formula. The circle $\displaystyle {(x-a)^{2}+(y-b)^{2}}={R^{2}}$ and the point $\displaystyle (x_0,y_0)$ on it:

the tangent’s equation by the slopes: $\displaystyle {(y_0-b)(y-y_0)}={-(x_0-a)(x-x_0)}$ .
Write $\displaystyle {x-x_0}={(x-a)-(x_0-a)}$ and $\displaystyle {y-y_0}={(y-b)-(y_0-b)}$ , and open the brackets.
$\displaystyle (x_0-a)^{2}+(y_0-b)^{2}$ appears, and it is $\displaystyle R^{2}$ , because the point is on the circle.

\displaystyle (x_0-a)(x-a)+(y_0-b)(y-b)=R^{2}

In the same example: $\displaystyle {(5-2)(x-2)+(3+1)(y+1)}={25}$ , that is $\displaystyle {3x-6+4y+4}={25}$ , and again $\displaystyle {3x+4y}={27}$ .

The formula works also when the tangent is parallel to the $\displaystyle y$ -axis and has no slope: in this circle, at the point $\displaystyle (7,-1)$ , it gives $\displaystyle {5(x-2)}={25}$ , that is $\displaystyle {x}={7}$ .

A tangent that moves with the circle

When a circle is shifted, its tangent moves with it by the same shift.

The tangent to the circle $\displaystyle {x^{2}+y^{2}}={25}$ at the point $\displaystyle (3,4)$ is $\displaystyle {3x+4y}={25}$ .
Shift the circle $\displaystyle 2$ to the right: $\displaystyle {(x-2)^{2}+y^{2}}={25}$ , and the point of contact moves to $\displaystyle (5,4)$ .
The tangent moves $\displaystyle 2$ to the right too: $\displaystyle x$ is replaced by $\displaystyle x-2$ , giving $\displaystyle {3(x-2)+4y}={25}$ , that is $\displaystyle {3x+4y}={31}$ .

Exercise 1

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

\displaystyle x^{2}+y^{2}=25

\displaystyle (-3,4)

Exercise 2

The point $\displaystyle (2,3)$ is on the circle $\displaystyle {(x+1)^{2}+(y-2)^{2}}={10}$ .
Write the equation of the tangent to the circle at this point.

\displaystyle (x+1)^{2}+(y-2)^{2}=10

\displaystyle (2,3)

Exercise 3

There is no need to find

\displaystyle a

: how did the circle and the point of contact move?

The tangent to the circle $\displaystyle {(x+a)^{2}+(y-4)^{2}}={25}$ at the point $\displaystyle (1,8)$ is $\displaystyle {4x+3y}={35}$ .
Write the equation of the tangent to the circle $\displaystyle {(x+a)^{2}+(y-6)^{2}}={25}$ at the point $\displaystyle (1,10)$ .

\displaystyle (1,10)

Exercise 4

Complete the square to find the centre and

\displaystyle R^{2}

The point $\displaystyle (6,3)$ is on the circle $\displaystyle {x^{2}+y^{2}-6x+2y-15}={0}$ .
Write the equation of the tangent at this point.

\displaystyle x^{2}+y^{2}-6x+2y-15=0

\displaystyle (6,3)

Exercise 5

A tangent of slope

\displaystyle 2

\displaystyle {y}={2x+n}

, and its distance from the centre equals the radius.

Find the tangents to the circle $\displaystyle {x^{2}+y^{2}}={20}$ whose slope is $\displaystyle 2$ .

An equation for

\displaystyle n

Write the two tangents on the board.

\displaystyle x^{2}+y^{2}=20

Exercise 6

The tangents to the circle $\displaystyle {x^{2}+y^{2}}={25}$ at the points $\displaystyle (3,4)$ and $\displaystyle (3,-4)$ meet at the point $\displaystyle T$ .

Write the equations of the two tangents.

Find

\displaystyle T

Formulas: analytic geometry, grade 12 (5 units)

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Points and lines

The distance between the points $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \sqrt{(x_1-x_2)^{2}+(y_1-y_2)^{2}}$
The slope of the line through the two points
$\displaystyle m$ $\displaystyle =$ $\displaystyle \frac{y_2-y_1}{x_2-x_1}$ $\displaystyle x_1\neq x_2$
The line with slope $\displaystyle m$ through $\displaystyle (x_1,y_1)$
$\displaystyle y-y_1=m(x-x_1)$
Perpendicular lines – $\displaystyle m_1$ , $\displaystyle m_2$ their slopes
$\displaystyle m_1\cdot m_2$ $\displaystyle =$ $\displaystyle -1$
The point $\displaystyle C$ dividing the segment $\displaystyle AB$ in the ratio $\displaystyle {AC:CB}={k:l}$ – $\displaystyle A(x_1,y_1)$ , $\displaystyle B(x_2,y_2)$
$\displaystyle C\left(\frac{lx_1+kx_2}{k+l},\frac{ly_1+ky_2}{k+l}\right)$
The distance of the point $\displaystyle (x_0,y_0)$ from the line $\displaystyle {Ax+By+C}={0}$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \frac{|Ax_0+By_0+C|}{\sqrt{A^{2}+B^{2}}}$
The distance between the parallel lines $\displaystyle {Ax+By+C_1}={0}$ and $\displaystyle {Ax+By+C_2}={0}$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \frac{|C_1-C_2|}{\sqrt{A^{2}+B^{2}}}$

Circle

A circle with centre $\displaystyle (a,b)$ and radius $\displaystyle R$
$\displaystyle (x-a)^{2}+(y-b)^{2}=R^{2}$
The expanded equation – a circle when, after completing the square, the right side is positive
$\displaystyle x^{2}+y^{2}+Cx+Dy+E=0$
The tangent to the circle at the point $\displaystyle (x_0,y_0)$ on it
$\displaystyle (x_0-a)(x-a)+(y_0-b)(y-b)=R^{2}$

Parabola

The canonical parabola
$\displaystyle y^{2}=2px$
The focus
$\displaystyle F\left(\frac{p}{2},0\right)$
The directrix
$\displaystyle x=-\frac{p}{2}$
The tangent at the point $\displaystyle (x_0,y_0)$ on the parabola
$\displaystyle y\cdot y_0=p(x+x_0)$

Ellipse

The canonical ellipse – the foci on the $\displaystyle x$ -axis when $\displaystyle a>b$
$\displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$
The distance of a focus from the origin
$\displaystyle c$ $\displaystyle =$ $\displaystyle \sqrt{a^{2}-b^{2}}$
The sum of the distances of a point $\displaystyle P$ on the ellipse from the foci
$\displaystyle PF_1+PF_2$ $\displaystyle =$ $\displaystyle 2a$

Hyperbola

The canonical hyperbola – the foci on the $\displaystyle x$ -axis
$\displaystyle \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
The distance of a focus from the origin
$\displaystyle c$ $\displaystyle =$ $\displaystyle \sqrt{a^{2}+b^{2}}$
The difference of the distances of a point $\displaystyle P$ on the hyperbola from the foci
$\displaystyle |PF_1-PF_2|$ $\displaystyle =$ $\displaystyle 2a$
The asymptotes
$\displaystyle y=\pm\frac{b}{a}x$