Equation of tangent from any point outside the circle:

METHOD 1:

Let the equation of given circle be (x - a)² + (y - b)² = r², and we have to find equation of tangent from the point P(x₁, y₁) outside the circle.

Equation of any line passing through P(x₁, y₁) is given by y - y₁ = m (x - x₁). For different the values of m we get different lines. For some specific value of m this equation will represent the tangent to the circle. To find the value of m apply the condition of tangency.

METHOD 2:

The joint equation of a pair of tangents drawn from the point P(x₁, y₁)  to the

circle x² + y² + 2gx + 2fy + c = 0 is T²=SS_1, here S º x²+y²+2gx +2fy + c = 0 ,S₁º x₁²+y₁²+2gx₁+2fy₁+c and T º xx₁+yy₁+g(x+x₁)+f(y +y₁)+c

Illustration: Find the equation of tangent. Draw to the circle x² + y² - 2x - 2y - 2 = 0  from point (4, 5)

Common Mistake:  Student usually use formula that equation of tangent at     (x₁, y₁) on circle   x² + y² + 2gx + 2fy + c = 0 is xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0. But they forget that in this case (x₁, y₁)  is lying on circle.

Solution:             Clearly (4, 5) is lying outside the given circle. Thus from the point (4, 5) we can draw 2 tangents. Equation of any line passing through (5, 4) is   y - 4 = m (x - 5).Now to find the value of m we will apply condition of tangency.

                                                Put these values in the equation (1) to find required tangency.

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