Finding coordinates of a point on a circle( angle and distance from O known)

[Wikeda]

1. I basically have to find the coordinates of P. All the pink lines are know, coordinates of points A and centre of circle are know.


2. (x-a)^2 + (y-b)^2 = r^2


3. I try to substitute mx+c into the equation and get

(x-a)^2 + (y-mx-c)^2 + r^2= 0

but I can't work out what m and c are. Any help would be appreciated! Am I in the right direction, or am I completely off?

 

[Mentallic]

1. I basically have to find the coordinates of P. All the pink lines are know, coordinates of points A and centre of circle are know.2. (x-a)^2 + (y-b)^2 = r^23. I try to substitute mx+c into the equation and get

(x-a)^2 + (y-mx-c)^2 + r^2= 0

but I can't work out what m and c are. Any help would be appreciated! Am I in the right direction, or am I completely off?

I'm just going off what it looks like on the picture, but it looks as though P is the intersection between a line with gradient -1 going through the centre of the circle and the circle itself. Is this correct?

If so, what is the equation of a line with gradient m that goes through the point (a,b) ?

EDIT: And by the way, if you were trying to find where a general circle intersects with a general line, then you need to plug
$$y=mx+c$$ into the equation
$$(x-a)^2+(y-b)^2=r^2$$
to get
$$(x-a)^2+(mx+c-b)^2=r^2$$
And then you'll have an equation that you have a quadratic in x that you have to solve (all the other values are constants, so the answer will depend on what those are).
Once you've done that, you can plug that value of x back into the line equation to find y (you can also plug into the circle equation but it's harder work to solve for y there and you also get two values).

 

[Wikeda]

Thank you for your response!
Actually The point P could be anywhere on the arch that is being formed by the circle, so the line doesn't intersects the centre of the circle.

I see now where I got it wrong, should have substituted y instead of b.

Thanks!

 

[Mentallic]

Thank you for your response!
Actually The point P could be anywhere on the arch that is being formed by the circle, so the line doesn't intersects the centre of the circle.

I see now where I got it wrong, should have substituted y instead of b.

Thanks!

Well then, if you knew the equation of the circle, you won't know P exactly because it obviously will change depending on how high up the circle it is. The equations are still correct though