A level Circles Maths Questions

I'm struggling with two questions.

Q1) O is the centre of a circle. The point P (2,4) lies on the circumference of the circle. What is the gradient of the tangent at P?

Q2) The line y=2x does not meet the circle (x-2)^2 + (y-1)^2=d. Find the range of possible values for d.

1. Co ordinate of O?
2. Substitute y=2x into circle eqn

We aren't given the co-ordinate of O and thank you

I'm struggling with two questions.

Q1) O is the centre of a circle. The point P (2,4) lies on the circumference of the circle. What is the gradient of the tangent at P?

Q2) The line y=2x does not meet the circle (x-2)^2 + (y-1)^2=d. Find the range of possible values for d.

The origin is (0,0). So you need to work out the gradient of the radius OP (change in y/ change in x). Tangents meet radii at 90 degrees therefore the gradient of the tangent is (-1)/ gradient (negative reciprocal).

The centre of that circle is (2,1). You need to work out a radius that makes sure that the line y=2x does not touch/ intercept the circle.

1. Co ordinate of O?
2. Substitute y=2x into circle eqn

Does that mean I'm allowed to assume that the co-ordinates of O are (0,0)? It doesn't say it's the origin.

I subbed y=2x into the circle equation and d=5x^2 -8x+5. How do I find the range of values though? Do I have to make sure b^2-4ac<0? because the discriminant is 36 but this is obviously larger than 0?

Good so far.
Now bring the d over to the other side so that your quadratic is equal to zero.
Now you can use the discriminant to find the range of values for d.