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 (x_{1}, y_{1}), we have to do the following replacements for x^{2}, y^{2}, x and y in the equation of the curve.

x^{2} ---> xx_{1}

y^{2} -----> yy_{1}

x -----> (x + x_{1})/2

y -----> (y + y_{1})/2

**Example 1 :**

Find the equation of a tangent to the circle

x^{2} + y^{2} = 25

at (4, 3).

**Solution :**

x^{2} + y^{2} = 25

Equation of the tangent to the above circle at (x_{1}, y_{1}) :

xx_{1} + yy_{1} = 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

x^{2} + y^{2} - 4x -3y + 12 = 0

at (2, 3).

**Solution:**

x^{2} + y^{2} - 4x -6y + 12 = 0

Equation of the tangent to the above circle at (x_{1}, y_{1}) :

xx_{1} + yy_{1} - 4 [(x+x_{1})/2] - 6 [(y +y_{1})/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

x^{2} + y^{2}- 4x -4y -8 = 0

at (-2, -2).

**Solution :**

x^{2} + y^{2} - 4x -4y -8 = 0

Equation of the tangent to the above circle at (x_{1}, y_{1}) :

xx_{1} + yy_{1} - 4 [(x+x_{1})/2] - 4 [(y+y_{1})/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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