Step-by-Step Guidance for Conic Sections: Parabolas, Ellipses, and Hyperbolas

Q1a. Find all the characteristics of the parabola described by the equation: $y - 32 = -16x + 1$

Background

Topic: Parabolas (Conic Sections)

This question tests your ability to identify the key features of a parabola from its equation, including the vertex, focus, axis of symmetry, directrix, and latus rectum.

Key Terms and Formulas

• Vertex: The turning point of the parabola.

• Focus: A fixed point used to define the parabola.

• Axis of Symmetry: The line that divides the parabola into two mirror images.

• Directrix: A fixed line used in the definition of the parabola.

• Latus Rectum: A line segment perpendicular to the axis of symmetry, passing through the focus, whose endpoints lie on the parabola.

Standard form for a parabola that opens horizontally: $(y - k)^2 = 4p(x - h)$

Step-by-Step Guidance

1. Rewrite the given equation $y - 32 = -16x + 1$ in standard form. Start by moving all terms to one side and grouping $x$ and $y$ terms appropriately.

2. Identify the values of $h$, $k$, and $p$ by comparing your equation to the standard form $(y - k)^2 = 4p(x - h)$.

3. Use the values of $h$, $k$, and $p$ to determine the vertex $(h, k)$ of the parabola.

4. Recall that the focus is located at $(h + p, k)$ for a horizontally opening parabola. Set up the calculation for the focus.

5. Write the equation for the axis of symmetry and set up the equation for the directrix, which is $x = h - p$ for this orientation.

Try solving on your own before revealing the answer!

Final Answer:

Standard form: $(y - 32)^2 = -16(x - 1)$

• Vertex: $(1, 32)$

• Focus: $(-3, 32)$

• Axis of symmetry: $y = 32$

• Directrix: $x = 5$

• Latus rectum: Length $= 16$

The parabola opens to the left. The latus rectum is a segment through the focus, perpendicular to the axis of symmetry, with endpoints at $(-3, 32 7 8)$.

Q1b. Find all the characteristics of the parabola described by the equation: $x^2 - 4x - 4y - 4 = 0$

Background

Topic: Parabolas (Conic Sections)

This question asks you to extract the vertex, focus, axis of symmetry, directrix, and latus rectum from a quadratic equation in $x$ and $y$.

Key Terms and Formulas

• Standard form for a vertical parabola: $(x - h)^2 = 4p(y - k)$

• Vertex: $(h, k)$

• Focus: $(h, k + p)$

• Axis of symmetry: $x = h$

• Directrix: $y = k - p$

• Latus rectum: $|4p|$

Step-by-Step Guidance

1. Rewrite the equation $x^2 - 4x - 4y - 4 = 0$ in the form $(x - h)^2 = 4p(y - k)$ by completing the square for $x$.

2. Identify $h$, $k$, and $p$ from the standard form.

3. Determine the vertex $(h, k)$.

4. Set up the calculation for the focus using $(h, k + p)$.

5. Write the equations for the axis of symmetry and the directrix, and set up the calculation for the latus rectum.

Try solving on your own before revealing the answer!

Final Answer:

Standard form: $(x - 2)^2 = 4(y + 1)$

• Vertex: $(2, -1)$

• Focus: $(2, 0)$

• Axis of symmetry: $x = 2$

• Directrix: $y = -2$

• Latus rectum: Length $= 4$

The parabola opens upward. The latus rectum passes through the focus at $(2 7 2, 0)$.

Q2a. Find the standard form equation of the parabola with the given characteristics: Vertex at $(1, -3)$; Focus at $(1, -1)$

Background

Topic: Parabolas (Conic Sections)

This question tests your ability to write the equation of a parabola given its vertex and focus.

Key Terms and Formulas

• Standard form for a vertical parabola: $(x - h)^2 = 4p(y - k)$

• Vertex: $(h, k)$

• Focus: $(h, k + p)$

• $p$ is the distance from the vertex to the focus.

Step-by-Step Guidance

1. Identify $h$ and $k$ from the vertex $(1, -3)$.

2. Find $p$ by calculating the vertical distance between the vertex and the focus.

3. Plug $h$, $k$, and $p$ into the standard form $(x - h)^2 = 4p(y - k)$.

4. Simplify the equation to obtain the standard form.

Try solving on your own before revealing the answer!

Final Answer:

Standard form: $(x - 1)^2 = 8(y + 3)$

The parabola opens upward, with vertex at $(1, -3)$ and focus at $(1, -1)$.

Q3a. Find all the characteristics of the ellipse described by the equation: $y - 229 + x + 124 = 1$

Background

Topic: Ellipses (Conic Sections)

This question asks you to identify the center, vertices, foci, axes, and intersection points of an ellipse from its equation.

Key Terms and Formulas

• Standard form for an ellipse: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

• Center: $(h, k)$

• Vertices: $(h 7 a, k)$ or $(h, k 7 b)$

• Foci: $(h 7 c, k)$ or $(h, k 7 c)$, where $c^2 = a^2 - b^2$

• Major axis: The longer axis of the ellipse

• Minor axis: The shorter axis of the ellipse

Step-by-Step Guidance

1. Rewrite the equation $y - 229 + x + 124 = 1$ in the standard form for an ellipse.

2. Identify the center $(h, k)$ by comparing to the standard form.

3. Determine $a$ and $b$ (the lengths of the semi-axes).

4. Set up the calculations for the vertices and foci using $a$, $b$, and $c$.

5. Write the equations for the major and minor axes, and set up the calculation for the intersection points with the minor axis.

Try solving on your own before revealing the answer!

Final Answer:

Standard form: $(x + 124) + (y - 229) = 1$ (This equation appears to be linear, not an ellipse. Please check for a possible typo.)

If this is meant to be an ellipse, please provide the correct equation.

Q3b. Find all the characteristics of the ellipse described by the equation: $4x^2 + 9y^2 - 16x - 20 = 0$

Background

Topic: Ellipses (Conic Sections)

This question tests your ability to extract the center, vertices, foci, and axes from a general quadratic equation of an ellipse.

Key Terms and Formulas

• Standard form: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

• Center: $(h, k)$

• Vertices: $(h 7 a, k)$ or $(h, k 7 b)$

• Foci: $(h 7 c, k)$ or $(h, k 7 c)$, where $c^2 = a^2 - b^2$

Step-by-Step Guidance

1. Rewrite the equation $4x^2 + 9y^2 - 16x - 20 = 0$ in standard form by completing the square for $x$ and $y$.

2. Identify $h$ and $k$ (the center) from the completed square form.

3. Determine $a$ and $b$ (the lengths of the semi-axes).

4. Set up the calculations for the vertices and foci using $a$, $b$, and $c$.

5. Write the equations for the major and minor axes, and set up the calculation for the intersection points with the minor axis.

Try solving on your own before revealing the answer!

Final Answer:

Standard form: $\frac{(x - 2)^2}{9} + \frac{y^2}{4} = 1$

• Center: $(2, 0)$

• Vertices: $(2 7 3, 0) = (5, 0)$ and $(-1, 0)$

• Foci: $(2 7 \sqrt{5}, 0)$

• Major axis: $y = 0$

• Minor axis: $x = 2$

• Intersection points with minor axis: $(2, 2)$ and $(2, -2)$

Q4a. Find the standard form equation of the ellipse with the given characteristics: Center at $(1, -2)$; a vertex at $(1, 0)$; minor axis intersection point at $(0, -2)$

Background

Topic: Ellipses (Conic Sections)

This question asks you to write the equation of an ellipse given its center, a vertex, and a point on the minor axis.

Key Terms and Formulas

• Standard form: $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$ (if major axis is vertical)

• Center: $(h, k)$

• Vertices: $(h, k 7 a)$ or $(h 7 b, k)$

• Minor axis intersection: $(h 7 b, k)$ or $(h, k 7 b)$

Step-by-Step Guidance

1. Identify $h$ and $k$ from the center $(1, -2)$.

2. Use the vertex $(1, 0)$ to find $a$ (distance from center to vertex along the major axis).

3. Use the minor axis intersection point $(0, -2)$ to find $b$ (distance from center to minor axis intersection).

4. Plug $h$, $k$, $a$, and $b$ into the standard form of the ellipse.

5. Simplify the equation to obtain the standard form.

Try solving on your own before revealing the answer!

Final Answer:

Standard form: $\frac{(x - 1)^2}{1^2} + \frac{(y + 2)^2}{2^2} = 1$

The ellipse is centered at $(1, -2)$, with a vertical major axis of length $4$and a horizontal minor axis of length $2$.

Q5a. Find all the characteristics of the hyperbola described by the equation: $x - 225 - y28 = 1$

Background

Topic: Hyperbolas (Conic Sections)

This question asks you to extract the center, vertices, foci, axes, and asymptotes from a hyperbola equation.

Key Terms and Formulas

• Standard form for a hyperbola: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ (horizontal transverse axis)

• Center: $(h, k)$

• Vertices: $(h 7 a, k)$

• Foci: $(h 7 c, k)$, where $c^2 = a^2 + b^2$

• Asymptotes: $y - k = 7 \frac{b}{a}(x - h)$

Step-by-Step Guidance

1. Rewrite the equation $x - 225 - y28 = 1$ in standard form for a hyperbola.

2. Identify $h$, $k$, $a$, and $b$ by comparing to the standard form.

3. Set up the calculations for the center, vertices, and foci.

4. Write the equations for the transverse and conjugate axes, and set up the equations for the asymptotes.

Try solving on your own before revealing the answer!

Final Answer:

Standard form: $\frac{(x - 225)}{1} - \frac{(y28)}{1} = 1$ (This equation appears to have a typo. Please check the equation for accuracy.)

If you provide the correct equation, I can give the full characteristics.

Q5b. Find all the characteristics of the hyperbola described by the equation: $5x^2 - 4y^2 + 10x + 8y - 19 = 0$

Background

Topic: Hyperbolas (Conic Sections)

This question tests your ability to extract the center, vertices, foci, axes, and asymptotes from a general quadratic equation of a hyperbola.

Key Terms and Formulas

• Standard form: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$

• Center: $(h, k)$

• Vertices: $(h 7 a, k)$

• Foci: $(h 7 c, k)$, where $c^2 = a^2 + b^2$

• Asymptotes: $y - k = 7 \frac{b}{a}(x - h)$

Step-by-Step Guidance

1. Rewrite the equation $5x^2 - 4y^2 + 10x + 8y - 19 = 0$ in standard form by completing the square for $x$ and $y$.

2. Identify $h$, $k$, $a$, and $b$ from the completed square form.

3. Set up the calculations for the center, vertices, and foci.

4. Write the equations for the transverse and conjugate axes, and set up the equations for the asymptotes.

Try solving on your own before revealing the answer!

Final Answer:

Standard form: $\frac{(x + 1)^2}{3} - \frac{(y - 1)^2}{5} = 1$

• Center: $(-1, 1)$

• Vertices: $(-1 7 \sqrt{3}, 1)$

• Foci: $(-1 7 \sqrt{8}, 1)$

• Transverse axis: $y = 1$

• Conjugate axis: $x = -1$

• Asymptotes: $y - 1 = 7 \sqrt{\frac{5}{3}}(x + 1)$

Q6a. Find the standard form equation of the hyperbola with the given characteristics: Center at $(1, -2)$; a focus at $(4, -2)$; a vertex at $(3, -2)$

Background

Topic: Hyperbolas (Conic Sections)

This question asks you to write the equation of a hyperbola given its center, a focus, and a vertex.

Key Terms and Formulas

• Standard form: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ (horizontal transverse axis)

• Center: $(h, k)$

• Vertices: $(h 7 a, k)$

• Foci: $(h 7 c, k)$, where $c^2 = a^2 + b^2$

Step-by-Step Guidance

1. Identify $h$ and $k$ from the center $(1, -2)$.

2. Find $a$ by calculating the distance from the center to the vertex along the $x$-axis.

3. Find $c$ by calculating the distance from the center to the focus along the $x$-axis.

4. Use $c^2 = a^2 + b^2$ to solve for $b^2$.

5. Plug $h$, $k$, $a^2$, and $b^2$ into the standard form of the hyperbola.

Try solving on your own before revealing the answer!

Final Answer:

Standard form: $\frac{(x - 1)^2}{4} - \frac{(y + 2)^2}{5} = 1$

The hyperbola is centered at $(1, -2)$, with vertices at $(3, -2)$ and $(-1, -2)$, and foci at $(4, -2)$ and $(-2, -2)$.