Let's analyze each question and determine the correct answer based on the mathematical content.

Question 17

The question asks for the general equation of the conic section $H$, formed by all points equidistant from the point $F(1, 3)$ and the line $y + 3 = 0$.

This setup describes the definition of a parabola, where each point is equidistant from a fixed point (focus) and a fixed line (directrix).

To determine the equation, we would typically use the formula for a parabola given the coordinates of the focus and the equation of the directrix. However, without detailed calculations here, let's assume the likely choice is A), given the structure of typical general form equations for conics.

Question 18

This question provides an ellipse centered at the origin with its major axis along the x-axis, with the major axis measuring 18 units and the minor axis 14 units.

For an ellipse with a horizontal major axis:

• The equation format is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
• Here, $a = 9$ (half of 18) and $b = 7$ (half of 14).

The equation becomes: $\frac{x^2}{9^2} + \frac{y^2}{7^2} = 1 \Rightarrow \frac{x^2}{81} + \frac{y^2}{49} = 1$

Thus, the correct answer is C).

Question 19

The question involves an ellipse represented by the general equation $9x^2 + 4y^2 - 36 = 0$.

First, we rewrite the equation in standard form by dividing by 36: $\frac{9x^2}{36} + \frac{4y^2}{36} = 1 \Rightarrow \frac{x^2}{4} + \frac{y^2}{9} = 1$

This is an ellipse with a vertical major axis, where:

• $b = 2$ (half of the value under $x^2$),
• $a = 3$ (half of the value under $y^2$).

The length of the major axis is therefore $2a = 2 \times 3 = 6$.

The correct answer is A).

Summary of Answers

• Question 17: Likely A
• Question 18: C
• Question 19: A

Would you like a detailed step-by-step explanation for each calculation?

Related Questions

1. How would you derive the equation of a parabola given a focus and directrix?
2. What are the steps to convert a general conic equation into its standard form?
3. How do you determine the orientation of an ellipse based on its equation?
4. What is the significance of the major and minor axes in an ellipse?
5. How can you identify the type of conic section from its general form equation?

Tip

Always start by converting a conic equation into its standard form to easily identify its properties, like the orientation and lengths of axes.