Equations of Lines: Slope-Intercept and Two-Point Form

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What is the equation of the line passing through the points (1, -1) and (3, 5)?

3x - y + 4 = 0

Define the y-intercept of a line.

The y-intercept is the point where the line intersects the y-axis, with coordinates (0, c).

What is the slope-intercept form of a line?

The slope-intercept form is y = mx + c, where m is the slope and c is the y-intercept.

How is the equation of a line found using the two-point form?

y - y1 = (y2 - y1) / (x2 - x1) * (x - x1) Signup and view all the answers

What are the coordinates of the y-intercept of a line?

(0, c) Signup and view all the answers

Explain the concept of finding the equation of a line with slope and points.

The equation can be determined using the formula y - y1 = m(x - x1) with the given slope and point (x1, y1). Signup and view all the answers

Find the equation of the line equidistant from the parallel lines $9x + 6y - 7 = 0$ and $3x + 2y + 6 = 0$.

$9x + 6y + 15 = 0$ Signup and view all the answers

A ray of light passing through the point (1, 2) reflects on the x-axis at point A and passes through (5, 3). Find the coordinates of point A.

(3, 0) Signup and view all the answers

Prove that the product of the lengths of the perpendiculars drawn from the point $\left(\frac{1}{\cos\theta}, \frac{1}{\sin\theta}\right)$ to the line $x\cos\theta + y\sin\theta = 1$ is $b^2$.

$b^2$ Signup and view all the answers

A person wants to reach the path $6x - 7y + 8 = 0$ in the least time from the junction of paths $2x - 3y + 4 = 0$ and $3x + 4y - 5 = 0$. Find the equation of the path to follow.

$3x - 4y + 13 = 0$ Signup and view all the answers

Explain the formula for the slope $m$ of a non-vertical line passing through points $(x_1, y_1)$ and $(x_2, y_2)$.

The formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $x_1 \neq x_2$. Signup and view all the answers

What is the condition for two lines to be parallel? Provide the relationship involving their slopes.

Two lines are parallel if and only if their slopes are equal. Signup and view all the answers

State the condition for a point to lie on a horizontal line.

The ordinate of the point must be a constant value, either a or -a. Signup and view all the answers

What is the equation of a horizontal line if it is at a distance a from the x-axis?

The equation of the line is y = a or y = -a. Signup and view all the answers

Explain how to determine if a given point lies on a given line algebraically.

Construct an algebraic equation involving the variables x and y that is true when the point is on the line and false otherwise. Signup and view all the answers

If a line is horizontal and a distance a from the x-axis, what is the relationship between the points on the line and the ordinate value?

The ordinate values of all points on the line are either a or -a. Signup and view all the answers

What is the significance of the equation y = a or y = -a for a horizontal line?

It represents the horizontal line's equation and characterizes the points on the line. Signup and view all the answers

How does the equation y = a or y = -a help determine the position of a point on a horizontal line?

By checking if the ordinate of the point matches the constant value a or -a. Signup and view all the answers