Alright, let's work through the steps to find the equation of the line in standard form, given that it has [tex]\( x \)[/tex]-intercept [tex]\((-P, 0)\)[/tex] and [tex]\( y \)[/tex]-intercept [tex]\((0, R)\)[/tex].
1. Identify intercepts:
- [tex]\( x \)[/tex]-intercept: [tex]\((-P, 0)\)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\((0, R)\)[/tex]
2. Form of the line equation using intercepts:
- The intercept form of the line equation is given by:
[tex]\[
\frac{x}{a} + \frac{y}{b} = 1
\][/tex]
Where [tex]\( a \)[/tex] is the [tex]\( x \)[/tex]-intercept and [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
3. Substitute intercepts into the intercept form:
- Here, [tex]\( a = -P \)[/tex] and [tex]\( b = R \)[/tex]. So the equation becomes:
[tex]\[
\frac{x}{-P} + \frac{y}{R} = 1
\][/tex]
4. Clear the denominators by multiplying through by [tex]\(-PR\)[/tex]:
- Multiply the entire equation by [tex]\(-PR\)[/tex] to remove the fractions:
[tex]\[
-R \cdot \frac{x}{-P} + P \cdot \frac{y}{R} = -PR
\][/tex]
5. Simplify the equation:
- Simplify each term to get a standard form linear equation:
[tex]\[
(-R \cdot \frac{x}{-P}) + (P \cdot \frac{y}{R}) = -PR
\][/tex]
[tex]\[
\left( \frac{R}{P} \right) x + \left( \frac{P}{R} \right) y = -PR
\][/tex]
- Note that the terms simplify to:
[tex]\[
\frac{R}{P} \cdot x + \frac{P}{R} \cdot y = -PR
\][/tex]
[tex]\[
R x - P y = PR
\][/tex]
6. Rewrite in standard form:
- Transform and rearrange to get:
[tex]\[
Px - Ry = PR
\][/tex]
Therefore, the equation of the line in standard form is:
[tex]\[ P x - R y = P R \][/tex]
This matches the fourth option in the provided list:
[tex]\[ \boxed{P x - R y = P R} \][/tex]
