Parametric equation of tangent question involving a circle

Question

Given a circle centered at the origin with radius $5$, find the parametric equation of the tangent line to the circle at the point $(3, −4)$.

The final equation is supposed to be like this: $$(3,-4) + t(x,y).$$

The problem is that I don't know what are the values of $x$ and $y$. Help, please.

Answer 1

Guide:

The radius is perpedicular to the tangent line.

Can you find a solution to $$3x-4y=0?$$

Answer 2

Hint: Draw a radius from the origin to $(3, -4)$. This line must be perpendicular to the desired tangent line. So the two lines' corresponding direction vectors must have a dot product of zero.

Alternatively, recall that the slopes of perpendicular lines must have a product of $-1$.

Answer 3

x and y are co-ordinate points of the tangent line and their values depend on the given value for parameter (t). The parametric equation of the tangent line can be written as,

x = 3(1+t), y = -4(1-t)

The point (3,-4) represents co-ordinate point on the tangent line for t=0 Similarly, other co-ordinate points on the tangent line will be obtained for giving values for t = ....-3, -2 -1 etc., and 1, 2, 3,....etc