analytic geometrycircles
How to find the equation of the circle which touches $y$ axis at $(0,3)$ and cuts a chord of length $8$ on the $x$ axis?
It should look like this:
Since the circle touches $y$ axis at $(0,3)$, its center has $y$-coordinate $3$. So the equation of the circle is of the form $(x-r)^2+(y-3)^2=r^2$.
How can I proceed further by using the fact that the circle passes through $(a,0)$ and $(b,0)$ with $b-a=8$? Anyway this would be very long. Is there some alternative?
Lets say the equation of the circle is given by:$$(x-a)^2+(y-b)^2=r^2$$Make use of the fact that the circle passes through $(4,-3)$ and $(0,7)$ to form two equations.
You are also told that it touches the circle:$$x^{2} + y^{2} - 6x + 2y + 5 = 0$$at $(4,-3)$ which mean the tangents of both circles at this point must be equal. This gives you a third equation.
You now have three equations and three unknowns which you should be able to solve.
For each $P$ there will be two circles drawn through it that will touch both the axes and their centres will obviously lie on the line $y=x$ as shown:
As is evident from the figure, the common chord $PQ$ will always be the line segment joining the point $P$ to the point $Q$ which is the reflection of $P$ with respect to the line $y=x$ . Hence the coordinate of $Q\equiv(b,a)$ .
Now to maximise the length of the common chord, the chord $PQ$ should be a diameter of the circle passing through $P$ with it's centre at $\left(\dfrac{a+b}{2},\dfrac{a+b}{2}\right)$ and this circle must also touch both the axes. Can you find the answer now?
EDIT: The equation of the circle with PQ as its diamter is : $$(x-a)(x-b)+(y-a)(y-b)=0$$
If on solving this circle with the say, the X axis, if there are repeated roots, then it will touch said axis. so we get the condition: $$(a+b)^2=8ab$$
Best Answer
HINT:
For $x$ intercept $y=0$
Setting $y=0, x^2-2xr+9=0$
If the circle cuts $x$ axis at $(x_1,0);(x_2,0)$ where $x_1\ge x_2$
We have $x_1-x_2=8$ and $\displaystyle x_1+x_2=2r,x_1x_2=9$
Use $\displaystyle (x_1-x_2)^2=(x_1+x_2)^2-4x_1x_2$ to find $r$