Conic Section D83B95

1. **Problem:** Find the center, vertices, and foci of the conic section given by $$16x^2 + 25y^2 + 16x + 4 = 100y$$. 2. **Problem:** Find the equation of the circle with radius $5$ tangent to the line $$3x + 4y = 24$$ at the point $\left(2, \frac{9}{2}\right)$. 3. **Problem:** Find the equation of the ellipse with center $(1,3)$, vertex $(1,-1)$, and passing through the origin. 4. **Problem:** Find the equation of the parabola with vertical axis, latus rectum $4$, passing through points $(6,4)$ and $(-4,-1)$. 5. **Problem:** Find the equation of the hyperbola with asymptotes $$x + y = 1$$ and $$x - y = 1$$, passing through $(3,1)$. --- ### Step 1: Conic section $$16x^2 + 25y^2 + 16x + 4 = 100y$$ 1. Rewrite the equation grouping $x$ and $y$ terms: $$16x^2 + 16x + 25y^2 - 100y + 4 = 0$$ 2. Complete the square for $x$ and $y$: - For $x$: $$16(x^2 + x) = 16\left(x^2 + x + \left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right)^2\right) = 16\left((x + \frac{1}{2})^2 - \frac{1}{4}\right)$$ - For $y$: $$25(y^2 - 4y) = 25\left(y^2 - 4y + 4 - 4\right) = 25\left((y - 2)^2 - 4\right)$$ 3. Substitute back: $$16\left((x + \frac{1}{2})^2 - \frac{1}{4}\right) + 25\left((y - 2)^2 - 4\right) + 4 = 0$$ 4. Expand: $$16(x + \frac{1}{2})^2 - 4 + 25(y - 2)^2 - 100 + 4 = 0$$ 5. Combine constants: $$16(x + \frac{1}{2})^2 + 25(y - 2)^2 - 100 = 0$$ 6. Add $100$ to both sides: $$16(x + \frac{1}{2})^2 + 25(y - 2)^2 = 100$$ 7. Divide both sides by $100$: $$\frac{(x + \frac{1}{2})^2}{\frac{100}{16}} + \frac{(y - 2)^2}{\frac{100}{25}} = 1$$ 8. Simplify denominators: $$\frac{(x + \frac{1}{2})^2}{6.25} + \frac{(y - 2)^2}{4} = 1$$ 9. Identify center, $a^2$, and $b^2$: - Center: $$\left(-\frac{1}{2}, 2\right)$$ - $a^2 = 6.25$, $a = 2.5$ - $b^2 = 4$, $b = 2$ 10. Since $a^2 > b^2$, major axis is horizontal. 11. Vertices: $$\left(-\frac{1}{2} \pm 2.5, 2\right) = (2, 2) \text{ and } (-3, 2)$$ 12. Foci distance: $$c = \sqrt{a^2 - b^2} = \sqrt{6.25 - 4} = \sqrt{2.25} = 1.5$$ 13. Foci: $$\left(-\frac{1}{2} \pm 1.5, 2\right) = (1, 2) \text{ and } (-2, 2)$$ --- ### Final answer for 1: - Center: $$\left(-\frac{1}{2}, 2\right)$$ - Vertices: $(2, 2)$ and $(-3, 2)$ - Foci: $(1, 2)$ and $(-2, 2)$ --- Since the user asked multiple problems but per instructions only the first problem is solved completely.