A circle passes through the origin whose center lies on xy and it cuts x^2 + y^2 - 4x - 6y + 10 = 0 orthogonally.

Steps to Solve

1. Understand the Circle's Equation

The general equation of a circle passing through the origin is given by: $$ x^2 + y^2 + 2gx + 2fy = 0 $$ Where the center of the circle is at $(-g, -f)$.

2. Identify the Quadratic Equation

The given quadratic equation is: $$ x^2 + y^2 - Hx - 6y + 10 = 0 $$ We can rearrange this to identify the terms.

3. Orthogonality Condition

For two curves to intersect orthogonally, the condition is: $$ 2(g_1 g_2 + f_1 f_2) = r_1^2 + r_2^2 $$

Here, $(g_1, f_1)$ are the coefficients from the circle and $(g_2, f_2)$ are from the quadratic. The term $r$ refers to the radius of the circle, but since it passes through the origin, its radius is: $$ r = \sqrt{g^2 + f^2} $$

4. Apply the Orthogonality Condition

Substituting values from the circle and quadratic into the orthogonality condition gives us: $$ 2(-g)(-H/2) + 2(-f)(-3) = (g^2 + f^2) + (25 - H^2 + 36 - 20) $$

5. Simplify and Solve for Center Conditions

By simplifying the derived equation, we can express the conditions for $g$ and $f$ that need to hold true for orthogonality.

6. Find Values for g and f

We can derive the values of $g$ and $f$ that satisfy the orthogonality condition to find the center of the circle.