A triangle has two of its sides along the axes. Its 3rd side touches the circle $\displaystyle x^2 + y^2 - 2ax - 2ay + a^2 = 0$. Find the equation of locus of the circumcentre of the triangle.
I can't provide you with the equation of the locus but I can give you a few hints:
1.
$\displaystyle x^2 + y^2 - 2ax - 2ay + a^2 = 0~\implies~(x-a)^2+(y-a)^2=a^2$
The circle touches both coordinate axes.
2. You are dealing with a right triangle where the circumcentre is the midpoint of the hypotenuse.
3. If the tangentpoint is T(t, p) then the tangent has the equation:
$\displaystyle (x-a)(t-a)+(y-a)(p-a)=a^2$
4. If the tangent is parallel to the coordinate axes then there doesn't exist a circumcentre. Therefore the straight lines x = a and y = a must be asymptotes of the locus.
5. I've drawn a sketch: Triangles in blue, locus in red, asymptotes in green
Let the circumcentre of the triangle $\displaystyle P(x_{1},y_{1}).$Therefore the the three vertices of the triangle will be $\displaystyle O(0,0),A(2x_{1},0)$ and $\displaystyle B(0,2y_{1}).$The Incentre of the triangle will be obviously $\displaystyle I(a,a)$ as can be seen in the diagram supplied by earboth.
Thus
$\displaystyle
a=\frac{(0)(2\sqrt{x_{1}^2+y_{1}^2})+(0)(2x_{1})+( 2x_{1})(2y_{1})}{2x_{1}+2y_{1}+2\sqrt{x_{1}^2+y_{1 }^2}}
$
Recall that the coordinates of the incentre of the triangle having vertices as $\displaystyle A(x_{1},y_{1}),B(x_{2},y_{2}) $and $\displaystyle C(x_{3},y_{3}) $is given by
$\displaystyle
(\frac{ax_{1}+bx_{2}+cx_{3}}{a+b+c},\frac{ay_{1}+b y_{2}+cy_{3}}{a+b+c})
$
where $\displaystyle a=BC,b=CA,c=AB$
On simplification,we get
$\displaystyle
x_{1}+y_{1}+\sqrt{x_{1}^2+y_{1}^2}=\frac{2x_{1}y_{ 1}}{a}
$
Therefore the required locus is
$\displaystyle
x+y+\sqrt{x^2+y^2}=\frac{2xy}{a}
$
This is the equation of the curve which earboth has drawn in red.