# Tangent & Normal to a Circle

### Tangent and Normal to a Circle in Coordinate Geometry:

#### 1. Tangent to a Circle:

• A tangent to a circle is a line that touches the circle at exactly one point, termed the point of tangency.
• It is perpendicular to the radius at the point of contact.

#### 2. Equation of Tangent to a Circle:

• For a circle with center $\left(h,k\right)$ and radius $r$, the equation of the tangent at a point $\left({x}_{1},{y}_{1}\right)$ on the circle is: $\left(x-{x}_{1}\right)\left({x}_{1}-h\right)+\left(y-{y}_{1}\right)\left({y}_{1}-k\right)={r}^{2}$ where $\left({x}_{1},{y}_{1}\right)$ are the coordinates of the point of tangency.

#### 3. Slope of Tangent:

• The slope of the tangent to a circle at the point of tangency is equal to the negative reciprocal of the slope of the radius at that point.

#### 4. Normal to a Circle:

• The normal to a curve at a given point is a line perpendicular to the tangent at that point.
• For a circle, the normal at a point on the circle is the line passing through that point and the circle's center.

#### 5. Equation of Normal to a Circle:

• The equation of the normal to a circle at a point $\left({x}_{1},{y}_{1}\right)$ on the circle with center $\left(h,k\right)$ is given by: $\left(y-{y}_{1}\right)=-\frac{\left(x-{x}_{1}\right)}{\left({x}_{1}-h\right)}$

#### 6. Properties:

• Perpendicularity: The tangent and the radius at the point of tangency are perpendicular.
• Normal as Perpendicular Line: The normal to a circle at a point is perpendicular to the tangent at that point.
• Unique Point of Contact: A tangent and normal intersect the circle at only one point.

#### 7. Derivation:

• The equations of the tangent and normal are derived based on the slopes of the radius and perpendicular lines, respectively.

### Example:

Consider a circle with center $\left(3,4\right)$ and radius $5$.

#### 1. Equation of Tangent:

• Find the equation of the tangent to the circle at the point $\left(7,4\right)$ on the circle.

Solution:

Given: Center of the circle $=\left(3,4\right)$
Radius $=5$
Point on the circle $=\left(7,4\right)$

The equation of the tangent to the circle at a point $\left({x}_{1},{y}_{1}\right)$ on the circle with center $\left(h,k\right)$ and radius $r$ is: $\left(x-{x}_{1}\right)\left({x}_{1}-h\right)+\left(y-{y}_{1}\right)\left({y}_{1}-k\right)={r}^{2}$

Substituting the values: $\left(x-7\right)\left(7-3\right)+\left(y-4\right)\left(4-4\right)={5}^{2}$

$\left(x-7\right)\left(4\right)=25$

$4x-28=25$

$4x=53$

$x=\frac{53}{4}$

So, the equation of the tangent to the circle at $\left(7,4\right)$ is $x=\frac{53}{4}$.

#### 2. Equation of Normal:

• Find the equation of the normal to the circle at the point $\left(7,4\right)$ on the circle.

Solution:

The equation of the normal to a circle at a point $\left({x}_{1},{y}_{1}\right)$ on the circle with center $\left(h,k\right)$ is given by: $\left(y-{y}_{1}\right)=-\frac{\left(x-{x}_{1}\right)}{\left({x}_{1}-h\right)}$

Substituting the values: $\left(y-4\right)=-\frac{\left(x-7\right)}{\left(7-3\right)}$

$\left(y-4\right)=-\frac{\left(x-7\right)}{4}$

$4\left(y-4\right)=-\left(x-7\right)$

$4y-16=-x+7$

$x+4y=23$

So, the equation of the normal to the circle at $\left(7,4\right)$ is $x+4y=23$.