Two parabolas have the same focus at $(4, 3)$. The directrices of the parabolas are the x-axis and the y-axis, respectively. The parabolas intersect at points $A$ and $B$. We need to find the value of $(AB)^2$.

1. Problem Description

Two parabolas have the same focus at $(4, 3)$. The directrices of the parabolas are the x-axis and the y-axis, respectively. The parabolas intersect at points $A$ and $B$. We need to find the value of $(AB)^2$.

2. Solution Steps

Let $P(x, y)$ be a point on the parabola. The distance from $P$ to the focus $(4, 3)$ is equal to the distance from $P$ to the directrix.

For the first parabola, the focus is $(4, 3)$ and the directrix is the x-axis ($y = 0$).

The equation of the first parabola is:

$\sqrt{(x-4)^2 + (y-3)^2} = |y|$

$(x-4)^2 + (y-3)^2 = y^2$

$x^2 - 8x + 16 + y^2 - 6y + 9 = y^2$

$x^2 - 8x - 6y + 25 = 0$

$6y = x^2 - 8x + 25$

$y = \frac{1}{6}x^2 - \frac{4}{3}x + \frac{25}{6}$

For the second parabola, the focus is $(4, 3)$ and the directrix is the y-axis ($x = 0$).

The equation of the second parabola is:

$\sqrt{(x-4)^2 + (y-3)^2} = |x|$

$(x-4)^2 + (y-3)^2 = x^2$

$x^2 - 8x + 16 + y^2 - 6y + 9 = x^2$

$y^2 - 6y - 8x + 25 = 0$

$8x = y^2 - 6y + 25$

$x = \frac{1}{8}y^2 - \frac{3}{4}y + \frac{25}{8}$

To find the intersection points, we need to solve the system of equations:

$6y = x^2 - 8x + 25$

$8x = y^2 - 6y + 25$

Subtract the two equations:

$6y - 8x = x^2 - 8x + 25 - (y^2 - 6y + 25)$

$6y - 8x = x^2 - y^2 + 6y - 8x$

$x^2 - y^2 = 0$

$x^2 = y^2$

$x = y$ or $x = -y$

If $x = y$, then

$6x = x^2 - 8x + 25$

$x^2 - 14x + 25 = 0$

$x = \frac{14 \pm \sqrt{14^2 - 4(25)}}{2} = \frac{14 \pm \sqrt{196 - 100}}{2} = \frac{14 \pm \sqrt{96}}{2} = \frac{14 \pm 4\sqrt{6}}{2} = 7 \pm 2\sqrt{6}$

So the intersection points are $A(7 + 2\sqrt{6}, 7 + 2\sqrt{6})$ and $B(7 - 2\sqrt{6}, 7 - 2\sqrt{6})$.

If $x = -y$, then

$-6x = x^2 - 8x + 25$

$x^2 - 2x + 25 = 0$

The discriminant is $D = (-2)^2 - 4(25) = 4 - 100 = -96 < 0$, so there are no real solutions.

Therefore, the intersection points are $A(7 + 2\sqrt{6}, 7 + 2\sqrt{6})$ and $B(7 - 2\sqrt{6}, 7 - 2\sqrt{6})$.

$(AB)^2 = (x_A - x_B)^2 + (y_A - y_B)^2 = (7 + 2\sqrt{6} - (7 - 2\sqrt{6}))^2 + (7 + 2\sqrt{6} - (7 - 2\sqrt{6}))^2$

$(AB)^2 = (4\sqrt{6})^2 + (4\sqrt{6})^2 = 16(6) + 16(6) = 96 + 96 = 192$

3. Final Answer

192