Circle tangent to y=x with point in interior and chord on x+y=0

[atavistic]

Homework Statement

A circle touches the line y=x at a pont P such that OP = 4*2^1/2 i.e 4root2 , where O is the origin.The circle contains the point (-10,2) in its interior and the length of its chord on the line x+y=0 is 6root2.Determine the equation of the circle.


The attempt at a solution

OK as its clear that the chord is perpendicular to the tangent so its part of the normal line and hence the diameter.So the length of the radius of the circle is 3root2.

Secondly using under-root S1 = length of tangent drawn from (x1,y1) I got c= 32.

I can't proceed any further.What use is the internal point?

 

[EnumaElish]

I would just draw this on graph paper and see if that helps.

 

[HallsofIvy]

Homework Statement

A circle touches the line y=x at a pont P such that OP = 4*2^1/2 i.e 4root2 , where O is the origin.The circle contains the point (-10,2) in its interior and the length of its chord on the line x+y=0 is 6root2.Determine the equation of the circle.


The attempt at a solution

OK as its clear that the chord is perpendicular to the tangent so its part of the normal line and hence the diameter.So the length of the radius of the circle is 3root2.

Secondly using under-root S1 = length of tangent drawn from (x1,y1) I got c= 32.

I can't proceed any further.What use is the internal point?

The distance from (0,0) to (x,y) is $\sqrt{x^2+ y^2}$. You are saying that OP= $4\sqrt{2}$ so $x^2+ y^2= 32$. Since, in addition, the point is on the line y= x, $2x^2= 32$, $x^2= 16$, x= 4. P is the point (4,4). The circle passes through the point (4,4). If you could find the center of the circle, you could use that to find the radius and then write down the equation of the circle.

Why is it "clear that the chord is perpendicular to the tangent"? the only time a chord is perpendicular to the tangent to the circle is when the chord is a diameter! You appear to use that to conclude that the radius of the circle must be half the length of that chord, but I can see no reason that it should be "clear".