EQUATION OF THE TANGENT

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 to a curve at (x₁, y₁), we have to do the following replacements for x², y², x and y in the equation of the curve. 

x² ---> xx₁

y² -----> yy₁

x -----> (x + x₁)/2

y -----> (y + y₁)/2

Example 1 :

Find the equation of a tangent to the circle

x² + y²  =  25

at (4, 3).

Solution :

x² + y²  =  25

Equation of the tangent to the above circle at (x₁, y₁) :

xx₁ + yy₁  =  25

Equation of the tangent to the circle at (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:

x² + y² - 4x -6y + 12  =  0

Equation of the tangent to the above circle at (x₁, y₁) :

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

Equation of the tangent to the circle at (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 :

x² + y² - 4x -4y -8  =  0

Equation of the tangent to the above circle at (x₁, y₁) :

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

Equation of the tangent to the circle at (-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

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