Tangents and chords

To find the equations of tangents and chords related to a circle, we use the geometric properties of circles and lines. Let’s explore how to calculate these step by step.

Finding the Equation of a Tangent to a Circle

The tangent to a circle is a straight line that touches the circle at exactly one point. The tangent is perpendicular to the radius at the point of contact.

Step 1: Start with the Circle Equation and the Point of Contact

Consider the circle with the equation:

$$x^2+y^2=r^2$$

and a point of contact $P(x_1,y_1)$ on the circle.

Step 2: Find the Gradient of the Radius

The gradient of the radius from the circle’s centre (at $(0,0)$) to $P(x_1,y_1)$ is:

$$\text{Gradient of radius}=\frac{y_1-0}{x_1-0}=\frac{y_1}{x_1}$$

Step 3: Find the Gradient of the Tangent

The tangent is perpendicular to the radius, so its gradient is the negative reciprocal of the radius gradient:

$$\text{Gradient of tangent}=-\frac{x_1}{y_1}$$

Step 4: Use the Point-Gradient Formula

The equation of the tangent can be written using the point-gradient formula:

$$y-y_1=m(x-x_1)$$

Substitute $m=-\frac{x_1}{y_1}$:

$$y-y_1=-\frac{x_1}{y_1}(x-x_1)$$

Simplify this to get the tangent equation in a standard form.

Finding the Equation of a Chord Between Two Points

A chord is a straight line segment connecting two points on the circle.

Step 1: Start with the Circle Equation and Two Points

Consider the circle with the equation:

$$x^2+y^2=r^2$$

and two points $P(x_1,y_1)$ and $Q(x_2,y_2)$ on the circle.

Step 2: Find the Gradient of the Chord

The gradient of the chord is calculated as:

$$\text{Gradient of chord}=\frac{y_2-y_1}{x_2-x_1}$$

Step 3: Use the Point-Gradient Formula

The equation of the chord can be written using the point-gradient formula. Using $P(x_1,y_1)$ as a reference point:

$$y-y_1=m(x-x_1)$$

Substitute $m=\frac{y_2-y_1}{x_2-x_1}$:

$$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$$

Simplify to find the equation of the chord.

Examples

Tangent Example

Consider the circle $x^2+y^2=25$ and the point $P(3,4)$. The gradient of the radius is:

$$\frac{y_1}{x_1}=\frac{4}{3}$$

The tangent’s gradient is:

$$-\frac{x_1}{y_1}=-\frac{3}{4}$$

Using the point-gradient formula:

$$y-4=-\frac{3}{4}(x-3)$$

Simplify to get:

$$3x+4y-25=0$$

Chord Example

Consider the circle $x^2+y^2=25$ and points $P(3,4)$ and $Q(4,3)$. The gradient of the chord is:

$$\frac{y_2-y_1}{x_2-x_1}=\frac{3-4}{4-3}=-1$$

Using the point-gradient formula with $P(3,4)$:

$$y-4=-1(x-3)$$

Simplify to get:

$$x+y-7=0$$

Summary

• For tangents, use the negative reciprocal of the radius gradient.
• For chords, calculate the gradient between two points and use the point-gradient formula.
• Simplify your equations to standard forms.