Equation of tangents:

Definition of tangent:  The tangent at any point of the circle if defined to be straight lines which meet the circle at that particular point but being produced, does not cut it at all. The point where tangent meets the curve is called the point of contact.

• Equation of the tangent to x² + y² + 2gx + 2fy + c = 0 at (x₁ , y₁) is  given by xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0.
• The condition that the straight line y = mx + c is a tangent to the circle x² + y²= a² is c² = a² (1 + m²) and the point of contact is (-a²m/c, a²/c) i.e. y = mx ± a √1+m² is always a tangent to the circle x² + y² = a² whatever be the value of m.

Illustration: Find the locus of the point of intersection of tangents to the circle x² + y² =4 which are at right angle to each other.

Solution:             Let point of intersection be ( h , k) . Equation of any tangent

                                    y = mx + 2 √1+m² 

                                    This passes through ( h ,k)

                                    K - mh = 2 √1+m² 

                              => (h² - 4 ) m² - 2 khm + k² - 4 = 0. Roots of the equation will represent slope of the tangents drawn from point (h, k). Let the roots be m₁ and m₂. Now both tangents are perpendicular to each other

                              => m₁.m₂ = - 1 => k²-4/h²-4 => h² + k² = 8

Alternate solution

                                    From diagram it is clear that

                                    ACBP will be a square

                                    Hence CA² + AP² = CP²

                                    => h² + k² = 4 + 4

                                    Therefore the locus is x² + y² = 8

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