Coordinate Geometry - Different Forms of the Equation of a Line-advanced

Find the equation of a line passing through the point $(2, -3)$ with a slope of $4$.

Given: Point $(x_1, y_1) = (2, -3)$ Slope $m = 4$ Using the point-slope form: $y - y_1 = m(x - x_1)$ $y - (-3) = 4(x - 2)$ $y + 3 = 4x - 8$ $4x - y - 11 = 0$

We apply the point-slope formula by substituting the given coordinates and the slope, then simplify to the general form $ax + by + c = 0$.

Find the equation of the line passing through the points $A(1, 2)$ and $B(3, 8)$.

Given: $(x_1, y_1) = (1, 2)$ $(x_2, y_2) = (3, 8)$ First, find the slope $m$: $m = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3$ Now, use the point-slope form with point $(1, 2)$: $y - 2 = 3(x - 1)$ $y - 2 = 3x - 3$ $3x - y - 1 = 0$

First, we calculate the slope using the two-point slope formula. Then, we use one of the points and the calculated slope in the point-slope equation.

Convert the general equation $3x - 4y + 12 = 0$ into intercept form and find the intercepts on the axes.

Given equation: $3x - 4y + 12 = 0$ Move the constant term to the right side: $3x - 4y = -12$ Divide the entire equation by $-12$ to make the right side equal to $1$: $\frac{3x}{-12} - \frac{4y}{-12} = \frac{-12}{-12}$ $\frac{x}{-4} + \frac{y}{3} = 1$ Comparing with $\frac{x}{a} + \frac{y}{b} = 1$, we get: $x$-intercept $a = -4$ $y$-intercept $b = 3$

To convert to intercept form, we isolate the constant and divide by it so the right side is $1$. The denominators of $x$ and $y$ then represent the intercepts.

1. For a line parallel to the $x$-axis, the $y$-coordinate remains constant for all points on the line. Since it passes through $(3, 2)$, the equation is $y = 2$.
2. For a line parallel to the $y$-axis, the $x$-coordinate remains constant. Since it passes through $(3, 2)$, the equation is $x = 3$.

A horizontal line has a slope $m=0$. Substituting into $y - y_1 = m(x - x_1)$ gives $y - 2 = 0(x - 3) \Rightarrow y = 2$. A vertical line has an undefined slope and takes the form $x = \text{constant}$.

Let the $y$-intercept be $b$. Then the $x$-intercept is $a = 2b$. The intercept form is $\frac{x}{a} + \frac{y}{b} = 1$. Substituting $a = 2b$: $\frac{x}{2b} + \frac{y}{b} = 1$ Multiply by $2b$: $x + 2y = 2b$ Since it passes through $(4, 3)$: $4 + 2(3) = 2b$ $4 + 6 = 2b \Rightarrow 10 = 2b \Rightarrow b = 5$ Thus $a = 2(5) = 10$. The equation is $\frac{x}{10} + \frac{y}{5} = 1$, or $x + 2y = 10$.

We use the intercept form $\frac{x}{a} + \frac{y}{b} = 1$ and the given ratio $a=2b$ to reduce the equation to one unknown variable $b$, then solve using the given point.