Circle equations, tangent, slope forms?

Question

1) The point (4,3) lies on the circle x^2 + y^2 = 25

Determine the slope of the line tangent to the circle @ (4,3)

2) Use the slope from #1 to determine the equation of the tangent line

3) If (a,b) lies on the circle x^2 + y^2 = r^2, show that the tangent line to the circle at that point has an equation ax+ by = r^2

Answer 1

1.

Differentiating with respect to x:

2x + 2y(dy / dx) = 0

dy / dx = - x / y

When x = 4 and y = 3, dy / dx = - 4 / 3

2.

The tangent is:

y - 3 = - (4 / 3)(x - 4)

3(y - 3) = 4(4 - x)

3y - 9 = 16 - 4x

4x + 3y = 25.

3.

Differentiating as before:

2x + 2y(dy / dx) = 0

dy / dx = - x / y

Gradient at (a, b) is - a / b.

The tangent has equation:

y - b = - (a / b)(x - a)

b(y - b) = a(a - x)

by - b^2 = a^2 - ax

ax + by = a^2 + b^2

ax + by = r^2.