A triangle has two of its sides along the axes. Its 3rd side touches the circle . 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.
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:
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 Therefore the the three vertices of the triangle will be and The Incentre of the triangle will be obviously as can be seen in the diagram supplied by earboth.
Thus
Recall that the coordinates of the incentre of the triangle having vertices as and is given by
where
On simplification,we get
Therefore the required locus is
This is the equation of the curve which earboth has drawn in red.