pls provide me the following and with 1 example each:

1)formula of distance between two lines
2) locus of a parabola

Sure, here are the details with examples for each request:

### 1. Formula of Distance Between Two Lines

The distance \(d\) between two parallel lines \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) can be calculated using the formula:

\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

**Example:**

Find the distance between the lines \(3x + 4y + 5 = 0\) and \(3x + 4y - 7 = 0\).

Using the formula:

\[ d = \frac{|-7 - 5|}{\sqrt{3^2 + 4^2}} = \frac{|-12|}{\sqrt{9 + 16}} = \frac{12}{\sqrt{25}} = \frac{12}{5} = 2.4 \]

So, the distance between the lines is 2.4 units.

### 2. Locus of a Parabola

A parabola is a set of all points \((x, y)\) that are equidistant from a fixed point called the focus and a fixed line called the directrix. The standard form of a parabola with its vertex at the origin and axis along the \(x\)-axis is given by:

\[ y^2 = 4ax \]

**Example:**

For a parabola with focus at \( (3, 0) \) and directrix \( x = -3 \):

1. The vertex will be the midpoint between the focus and directrix, which is \( (0, 0) \).
2. The distance between the vertex and the focus (or the directrix) is \( 3 \), which is \(a\) in the standard equation.

Thus, the equation of the parabola is:

\[ y^2 = 4 \cdot 3 \cdot x \]
\[ y^2 = 12x \]

### Would you like more details or have any questions?

### Relative Questions:

1. What is the formula to find the distance between two non-parallel lines?
2. How does the distance formula change if the lines are not in the standard form \(Ax + By + C = 0\)?
3. Can you derive the formula for the distance between two lines from the general form?
4. What are the properties of a parabola in its standard form?
5. How does the orientation of the parabola change its equation?

### Tip:
For a quick verification of parallel lines, ensure their slopes are equal. For lines \(Ax + By + C = 0\) and \(Ax + By + D = 0\), the slope \(\frac{-A}{B}\) should be the same.

pls provide me the following and with 1 example each: 1)formula of distance between two lines 2) locus of a parabola

Explore the formula for calculating the distance between two parallel lines and the locus of a parabola. Includes examples and detailed explanations of how to apply these formulas. Perfect for high school students in grades 9-12.

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