1. How to solve this?

hello im bit confused how to solve this

1.Show that the line y = mx + c touches the circle x^2 + y^2 = a^2 if c^2 = a^2(1+m^2)

2.The point (p,q) lies on the circle x^2 + y^2 = a^2.
Show that the equation of the tangent to the circle at (p,q) is px + qy = a^2

3. A circle passes through the points C (4,4) and D (10,2).
The line 2y = x + 4 is a tangent to the circle at C
Find the euqation of the circle

thanks

2. 2. x^2 + y^2 = a^2

x + ydy/dx = 0

dy/dx = -x/y

at (p,q) dy/dx = -p/q

The tangent line is y = -p/q x + c

at (p,q) q = -p^2/q +C

q^2 + p^2 = Cq

a^2/q = C

y = -p/q x + a^2/q

qy = -px +a^2

result follows

3.

(x-h)^2 + (y-k)^2 = r^2

2(x-h) + 2 (y-k)dy/dx = 0

dy/dx = - (x-h)/(y-k)

at (4,4) dy/dx = -(4-h)/(4-k)

Since 2y = x + 4

-(4-h)/(4-k) = 1/2

2h -8 = 4 - k

k = -2h + 4

So

(x-h)^2 + (y-k)^2 = r^2

becomes

(x-h)^2 + (y +2h -4) ^2 = r^2

Use the points (4,4) and (10,2) in the above and you'll get an easy system of 2 eqs to solve for r and h and then k

3. make sure you use the edited version of my last post