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 (h,k) and radius r, the equation of the tangent at a point (x1,y1) on the circle is: (xx1)(x1h)+(yy1)(y1k)=r2 where (x1,y1) 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 (x1,y1) on the circle with center (h,k) is given by: (yy1)=(xx1)(x1h)

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 (3,4) and radius 5.

1. Equation of Tangent:

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

Solution:

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

The equation of the tangent to the circle at a point (x1,y1) on the circle with center (h,k) and radius r is: (xx1)(x1h)+(yy1)(y1k)=r2

Substituting the values: (x7)(73)+(y4)(44)=52

 (x7)(4)=25

 4x28=25

 4x=53

x=534

So, the equation of the tangent to the circle at (7,4) is x=534.

2. Equation of Normal:

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

Solution:

The equation of the normal to a circle at a point (x1,y1) on the circle with center (h,k) is given by: (yy1)=(xx1)(x1h)

Substituting the values: (y4)=(x7)(73)

 (y4)=(x7)4

4(y4)=(x7)

 4y16=x+7

 x+4y=23

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