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

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@HaleCesium

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?

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

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

<p>(0, c)</p> Signup and view all the answers

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

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

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

<p>$9x + 6y + 15 = 0$</p> 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.

<p>(3, 0)</p> 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 $b^2$.

<p>$b^2$</p> 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.

<p>$3x - 4y + 13 = 0$</p> 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 $(x_2, y_2)$.

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

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

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

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

<p>The ordinate of the point must be a constant value, either a or -a.</p> 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?

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

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

<p>Construct an algebraic equation involving the variables x and y that is true when the point is on the line and false otherwise.</p> 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?

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

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

<p>It represents the horizontal line's equation and characterizes the points on the line.</p> 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?

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

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