## Tangents and Normals

### From DT Online

## Contents

##### Description

**Tangents** are lines *just touching* a given curve and its **Normal** is a line perpendicular to it at the point of contact *(or point of Tangency)*.

**Circle**, the

**Normal**to a

**Tangent**from a point

**P**on the circumference is a line joining the point to the circle centre - and the

**Tangent**is at right angles to the

**Normal**. This forms the basis for methods of constructing

**Tangents**such that the

**Point of Tangency**is given accurately.

##### To construct a Tangent to a Circle from a point outside its circumference:

- Join the point to the circle centre.
- Draw a semi-circle on this line as the diameter.
- The Point of Contact (
*or Tangency)*for the Tangent is where the semi-cicle intersects the circle and its Normal joins this point to the circle centre*(angle in semi-circle)*.

##### Construction of Tangents to an Ellipse

**To construct a Tangent to an Ellipse from any point on the curve:**

- Join the point to each of the two Foci as shown and bisect the angle found between the two lines.
- The bisector is the Normal and a Tangent may be drawn at right angles to it.

**To construct a Tangent to an Ellipse from any point outside the curve:**

- Strike an arc with the point as centre and radius equal to its distance from the Focus as shown.
- Strike a second arc with the second Focus as centre and radius set to equal the Major Axis.
- Join the points of intersection of the two arcs to the second Focus as shown.
- Where the two lines intersect the curve give the two Points of Tangency.
- Join the point to these Points of Tangency to produce the two possible Tangents and their Normals may be drawn at right angles to them.

##### Construction of Tangents to a Parabola

**Three methods to construct a Tangent to a point on the Parabola**

**Method 1.**

- Join the point to the Focus and draw from it a perpendicular to the Directrix.
- The bisector of the two lines created is the Tangent and its Normal may be constructed at right angles to it.

**Method 2.**

- Draw a line from the point perpendicular to the Axis
*(i.e. an Ordinate)* - With the Vertex as centre and radius the distance from the Vertex to where the Ordinate intersects the Axis, draw a semi-circle.
- Draw the Tangent to the point from where the semi-circle cuts the Axis as shown and its Normal may be constructed at right angles to it.

**Method 3.**

- Join the point to the Focus construct a line at right angles to it from the Focus.
- Draw the Tangent to the point from the line's intersection the Directrix and its Normal may be constructed at right angles to it.

##### To construct a Tangent from a point outside the Parabola

- Join the point (P) to the Focus(F) and using this as diameter, decribe a circle.
- Draw Tangent at the Vertex (V) perpendicular to the Axis to intersect the circle circumference in X and Y.
- Join PX and PY and produce both to touch the Parabola generating the two Tangents as shown.
- To draw a Normal, produce a Tangent
*(e.g. YP)*to meet the Directrix in D and join D to the Focus(F). - A line drawn perpendicular to DF will intersect the Parabola at a Point of Tangency(T).
- Construct a perpendicular to the Tangent from its Point of Tangency(T) to create its Normal.

##### Other terms common to all conics

**Tangents** and **Normals** are two terms used to describe properties of all conics. Other common properties are as follows:

- A
**Chord**is a straight line joining two points on the curve. - A
**Focal Chord**is a Chord which passses through the**Focus** - The mid-points of parallel Chords lie in a straight line called a
**Diameter** - A perpendicular from a point on the
**Axis**is called an**Ordinate**and if it goes straight across to the other side, it is a**Double Ordinate**. - The Double Ordinate through the Focus is the
**Latus Rectum***(translates from Latin as ‘The Right Side’)*. - A
**Normal**to any conic is at right angles to a**Tangent**at the**Point of Contact**