In this page equation of the tangent we are going to see how to find the equation of a tangent to the circle.First let us see the definition.

A tangent to a circle is a straight line which intersects (touches) the circle exactly one point.We can draw only two tangents to a circle from the point outside to a circle.

To get the equation of a tangent at(x₁,y₁) we have to replace x² by xx₁, y² by yy₁ , x as (x + x₁)/2 and y as (y + y₁)/2 in the equation of a circle.

Here we are going to see the examples to find the equation of the tangent to the circle.

**Example 1:**

Find the equation of a tangent to the circle x² + y² = 25 at (4,3).

**Solution:**

As per the procedure, we have to replace x² by xx₁ and y² by yy₁.

Equation of a circle x² + y² = 25

Equation of the tangent

xx₁ + yy₁ = 25

Here (x₁,y₁) is (4,3)

x(4) + y(3) = 25

**4x + 3y = 25**

**Example 2:**

Find the equation of tangent to the circle x² + y² - 4x -3y +12 = 0 at (2,3).

**Solution:**

Equation of the circle x² + y² - 4x -6y +12 = 0

We have to replace x² by xx₁ y² by yy₁,x as (x+x₁)/2 and y as (y +y₁)/2

xx₁ + yy₁ - 4 {(x+x₁)/2} - 6 {(y +y₁)/2} + 12 = 0

Equation of a circle at a point (2,3)

x(2) + y(3) + 2(x +2) -3(y+ 3) + 12 = 0

2x + 3y + 2x + 4 - 3y - 9 + 12 = 0

4x + 16 - 9 = 0

** 4x + 7 = 0**

**Example 3:**

Find the equation of the tangent to x² + y² - 4x -4y -8 = 0 at (-2,-2).

**Solution:**

Equation of the circle x² + y² - 4x -4y -8 = 0

We have to replace x² by xx₁ y² by yy₁,x as (x+x₁)/2 and y as (y +y₁)/2

xx₁ + yy₁ - 4 {(x+x₁)/2} - 4 {(y +y₁)/2} - 8 = 0

Equation of a circle at a point (-2,-2)

x(-2) + y(-2) + 2(x -2) -2(y - 2) -8 = 0

-2x - 2y + 2x - 4 - 2y + 4 - 8 = 0

-4y - 8 = 0

4y + 8 = 0

** y + 2 = 0**

**Related Topics**

**Length of the tangent to a circle****Family of circles****Orthogonal circles****Section formula****Area of triangle****Area of triangle worksheets****Area of quadrilateral****Centroid****Centroid of the triangle worksheets****Finding missing vertex using centroid worksheet****Midpoint****Distance between two points****Distance between two points worksheet****Slope of the line****Equation of the line****Equation of line using two points worksheet****Equation of the line using point and slope worksheets****Point of intersection of two lines****Point of intersection Worksheets****concurrency of straight line****Concurrency of straight lines worksheet****Circumcentre of a triangle****Circumcentre of Triangle Worksheet****Orthocentre of a triangle****Orthocentre of Triangle Worksheet****Incentre of a triangle****Locus****Perpendicular distance****Angle between two straight lines****Equation of a circle****With center and radius****With endpoints of a diameter****Equation of a circle passing though three points****Parabola****Ellipse**

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