Analytic Geometry: Study Guide for Parabolas, Ellipses, and Circles

Properties and Analysis of Parabolas

• Standard Equation Forms for Parabolas:     - The vertical parabola (opening up or down) is defined by the standard form: $(x - h)^2 = 4p(y - k)$ or $y = \frac{1}{4p}(x - h)^2 + k$.     - The horizontal parabola (opening left or right) is defined by the standard form: $(y - k)^2 = 4p(x - h)$.     - In these equations, the Point $(h, k)$ represents the Vertex.     - The variable $p$ represents the distance from the vertex to the focus and from the vertex to the directrix.

• Specific Parabola Problem: $8y = (x - 1)^2$:     - Transformation to Standard Form: This can be written as $y = \frac{1}{8}(x - 1)^2$.     - Identifying the Vertex: Comparing with $(x - h)^2 = 4p(y - k)$, we see $h = 1$ and $k = 0$. Vertex: $(1, 0)$.     - Calculating p: Setting $4p = 8$, we solve for $p$ as $p = \frac{8}{4} = 2$.     - Identifying the Focus: For a vertical parabola opening upward, the focus is $(h, k + p)$. Substituting the values: $(1, 0 + 2) = (1, 2)$. Focus: $(1, 2)$.     - Identifying the Directrix: The directrix is a horizontal line defined by $y = k - p$. Substituting the values: $y = 0 - 2 = -2$. Directrix: $y = -2$.

• Specific Parabola Problem: $4y + x^2 = 0$:     - Transformation: Rewrite the equation as $x^2 = -4y$.     - Vertex: The vertex is at the origin since there are no shifts for $x$ or $y$. Vertex: $(0, 0)$.     - Calculating p: Setting $4p = -4$, we solve for $p$ as $p = -1$. This indicates the parabola opens downward.     - Focus: Calculated as $(h, k + p) = (0, 0 + (-1)) = (0, -1)$. Focus: $(0, -1)$.     - Directrix: Calculated as $y = k - p = 0 - (-1) = 1$. Directrix: $y = 1$.

• Specific Parabola Problem: $8y = x^2$:     - Vertex: $(0, 0)$.     - Calculating p: $4p = 8 \implies p = 2$.     - Focus: $(0, 2)$.     - Directrix: $y = -2$.

• Specific Parabola Problem: $y - 1 = (x - 2)^2$:     - Vertex: $(2, 1)$.     - Calculating p: Since the coefficient of the squared term is $1$, then $4p = 1$, which means $p = \frac{1}{4}$.     - Focus: $(h, k + p) = (2, 1 + 0.25) = (2, 1.25)$.     - Directrix: $y = k - p = 1 - 0.25 = 0.75$.

Properties and Analysis of Ellipses

• Standard Equation Forms for Ellipses:     - Center: $(h, k)$.     - Horizontal Orientation: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$, where $a > b$.     - Vertical Orientation: $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$, where $a > b$.     - Major Axis: Length is $2a$. Vertices are the endpoints of the major axis.     - Minor Axis: Length is $2b$. Co-vertices are the endpoints of the minor axis.     - Foci: Located on the major axis at distance $c$ from the center, where $c^2 = a^2 - b^2$.

• Ellipses Problem 1: $\frac{(x + 2)^2}{4} + \frac{(y - 5)^2}{36} = 1$:     - Center: Found at $(h, k) = (-2, 5)$.     - Orientation: Since the larger denominator ($36$) is under the $(y - 5)^2$ term, the orientation is Vertical.     - Identifying a and b: $a^2 = 36 \implies a = 6$ and $b^2 = 4 \implies b = 2$.     - Vertices (Major Axis Endpoints): Located at $(h, k \pm a) = (-2, 5 \pm 6)$. Vertices: $(-2, 11)$ and $(-2, -1)$.     - Co-Vertices (Minor Axis Endpoints): Located at $(h \pm b, k) = (-2 \pm 2, 5)$. Co-vertices: $(0, 5)$ and $(-4, 5)$.     - Foci: $c^2 = a^2 - b^2 = 36 - 4 = 32$. Therefore, $c = \sqrt{32} = 4\sqrt{2} \approx 5.66$. Foci: $(-2, 5 \pm 4\sqrt{2})$.

• Ellipses Problem 2: $\frac{(x + 5)^2}{16} + \frac{y^2}{49} = 1$:     - Center: $(-5, 0)$.     - Orientation: Vertical (as $49 > 16$ and is under the $y$ term).     - Values: $a^2 = 49 \implies a = 7$; $b^2 = 16 \implies b = 4$.     - Vertices: $(-5, 0 \pm 7) \implies (-5, 7)$ and $(-5, -7)$.     - Co-Vertices: $(-5 \pm 4, 0) \implies (-1, 0)$ and $(-9, 0)$.     - Foci: $c^2 = 49 - 16 = 33 \implies c = \sqrt{33}$. Foci: $(-5, \pm \sqrt{33})$.

Properties and Analysis of Circles

• Standard Equation of a Circle:     - The standard form is: $(x - h)^2 + (y - k)^2 = r^2$.     - Center: $(h, k)$.     - Radius: $r$.

• Writing Equations from Given Information:     1. Center: (16, -2), Radius: 3:        - $(x - 16)^2 + (y - (-2))^2 = 3^2$        - Standard form: $(x - 16)^2 + (y + 2)^2 = 9$.     2. Center: (2, -2), Radius: 5:        - $(x - 2)^2 + (y - (-2))^2 = 5^2$        - Standard form: $(x - 2)^2 + (y + 2)^2 = 25$.

• Identifying Center and Radius from Standard Form:     5. (x - 4)² + (y + 4)² = 9:        - Center: $(4, -4)$.        - Radius: $r = \sqrt{9} = 3$.     6. (x + 4)² + (y + 3)² = 4:        - Center: $(-4, -3)$.        - Radius: $r = \sqrt{4} = 2$.

• Converting General Form to Standard Form (Completing the Square):     15. $x^2 + y^2 - 14x + 30y + 270 = 0$:         - Group terms: $(x^2 - 14x) + (y^2 + 30y) = -270$         - Complete the square for $x$: $(14 / 2)^2 = 49$         - Complete the square for $y$: $(30 / 2)^2 = 225$         - Add to both sides: $(x^2 - 14x + 49) + (y^2 + 30y + 225) = -270 + 49 + 225$         - Standard form: $(x - 7)^2 + (y + 15)^2 = 4$         - Center: $(7, -15)$.         - Radius: $r = \sqrt{4} = 2$.          16. $x^2 + y^2 - 4x + 8y - 44 = 0$:         - Group terms: $(x^2 - 4x) + (y^2 + 8y) = 44$         - Complete the square for $x$: $(4 / 2)^2 = 4$         - Complete the square for $y$: $(8 / 2)^2 = 16$         - Add to both sides: $(x^2 - 4x + 4) + (y^2 + 8y + 16) = 44 + 4 + 16$         - Standard form: $(x - 2)^2 + (y + 4)^2 = 64$         - Center: $(2, -4)$.         - Radius: $r = \sqrt{64} = 8$.