The absolute difference between the squares of the radii of the two circles passing through the point (-9, 4)$(-9, 4)$ and touching the lines x + y = 3$x + y = 3$ and x - y = 3$x - y = 3$, is equal to ________.

### Related Formula Perpendicular distance from point (x_0, y_0)$(x_0, y_0)$ to line Ax + By + C = 0$Ax + By + C = 0$: d = frac|Ax_0 + By_0 + C|sqrtA^2 + B^2$$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$$ ### Core Logic Since the circle touches two symmetric intersecting lines, its center must lie on their angle bisector (x$x$-axis). Use this property to find the center parameters.

### Step 1: Establish Center and Radius Equations The lines are x+y-3=0$x+y-3=0$ and x-y-3=0$x-y-3=0$. The intersection point is (3,0)$(3,0)$, and the bisector line is the x$x$-axis. Let the center be C(a, 0)$C(a, 0)$. The radius r$r$ is the perpendicular distance to either line: r = frac|a - 0 - 3|sqrt1^2 + 1^2 = frac|a - 3|sqrt2$$r = \frac{|a - 0 - 3|}{\sqrt{1^2 + 1^2}} = \frac{|a - 3|}{\sqrt{2}}$$ ### Step 2: Apply Point Passage Constraint The circle equation is (x - a)^2 + y^2 = r^2$(x - a)^2 + y^2 = r^2$. Substitute the given passage point (-9, 4)$(-9, 4)$: (-9 - a)^2 + 4^2 = left(fraca - 3sqrt2right)^2$$(-9 - a)^2 + 4^2 = \left(\frac{a - 3}{\sqrt{2}}\right)^2$$ 2left(a^2 + 18a + 81 + 16right) = a^2 - 6a + 9$$2\left(a^2 + 18a + 81 + 16\right) = a^2 - 6a + 9$$ 2a^2 + 36a + 194 = a^2 - 6a + 9 implies a^2 + 42a + 185 = 0$$2a^2 + 36a + 194 = a^2 - 6a + 9 \implies a^2 + 42a + 185 = 0$$ ### Step 3: Solve for Quadratic Roots Factor the quadratic equation: (a + 37)(a + 5) = 0 implies a_1 = -37, quad a_2 = -5$$(a + 37)(a + 5) = 0 \implies a_1 = -37, \quad a_2 = -5$$ ### Step 4: Compute Radii Squares Difference Find the corresponding radius value for each root: r_1 = frac|-37 - 3|sqrt2 = frac40sqrt2 = 20sqrt2 implies r_1^2 = 800$$r_1 = \frac{|-37 - 3|}{\sqrt{2}} = \frac{40}{\sqrt{2}} = 20\sqrt{2} \implies r_1^2 = 800$$ r_2 = frac|-5 - 3|sqrt2 = frac8sqrt2 = 4sqrt2 implies r_2^2 = 32$$r_2 = \frac{|-5 - 3|}{\sqrt{2}} = \frac{8}{\sqrt{2}} = 4\sqrt{2} \implies r_2^2 = 32$$ The absolute difference between their squares is: left|r_1^2 - r_2^2right| = |800 - 32| = 768$$\left|r_1^2 - r_2^2\right| = |800 - 32| = 768$$ ### Pattern Recognition Recognizing that the center must lie on the line of symmetry (x$x$-axis) eliminates one variable parameter immediately, reducing a difficult geometric system to a simple single-variable quadratic equation. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Circles Class 11 Mathematics: Straight Lines