The point P, (-1,4) lies on a circle C that is centered about the origin. Find the equation of the tangent to the circle at point P.

The most important fact to remember with this question is that two perpendicular straight lines have gradients related by m₁ = -1/m_2. This can then be used to find the gradient of the tangent by first finding the gradient of the line passing through the origin and point P. This gradient, m₁ is given by m₁ = (4-0)/(-1-0) = -4. The gradient of the tangent is therefor m₂ = -1/(-4) = 1/4. The equation of a straight line can be expressed as y = mx + c, so we have y = (1/4)x + c. As the tangent passes through P, the coordinates of P can be used to find c: (4) = (1/4)(-1) + c => c = 4 + 1/4 = 17/4.This provides the equation of the tangent to be y = (1/4)x + (17/4)