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3 problems match “hyperbolas”. Clear

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1. Hyperbola through three points ●●○○○ 20 ptsA hyperbola is centered at the origin and opens either horizontally or vertically. It passes through the points $(-3, 4)$, $(-2, 0)$, and $(t, 2)$. Find $t^2$.
2. Rectangle area with vertices on a hyperbola ●●●○○ 50 ptsThe diagonals of a rectangle are both 10 and intersect at $(0,0)$. All four vertices of the rectangle lie on the curve $y=\frac{12}{x}$. Calculate the area of the rectangle.
3. Minimum of a hyperbola distance expression ●●●●○ 100 ptsLet complex numbers $z_1$ and $z_2$ satisfy $\operatorname{Re} z_1>0$, $\operatorname{Re} z_2>0$, and $\operatorname{Re}(z_1^2)=\operatorname{Re}(z_2^2)=2$. Find the minimum value of $|z_1+2|+|\overline{z_2}+2|-|\overline{z_1}-z_2|$. Round to two decimal places.