3. TWO-DIMENSIONAL COORDINATE GEOMETRY

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3. TWO-DIMENSIONAL COORDINATE GEOMETRY

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3.1 General Introduction, Distance Formula, and Section Formula

Coordinate Geometry deals with the study of geometric shapes and figures using coordinates on a plane. In two dimensions, the position of a point is defined using an ordered pair $(x, y)$, where $x$ is the horizontal distance and $y$ is the vertical distance.

Distance Formula:

The distance between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is given by the formula:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Example: Find the distance between the points $A(2, 3)$ and $B(5, 7)$:

$$d = \sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$

Section Formula:

The section formula is used to find the coordinates of a point $P$ dividing the line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m:n$.

The coordinates of point $P$ dividing the segment in the ratio $m:n$ are:

$$P\left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right)$$

Example: If point $P$ divides the segment joining $A(1,2)\; and$$B(4, 6)$ in the ratio $2:3$, the coordinates of $P$ are:

$$P = \left( \frac{2(4) + 3(1)}{2 + 3}, \frac{2(6) + 3(2)}{2 + 3} \right) = \left( \frac{8 + 3}{5}, \frac{12 + 6}{5} \right) = \left( \frac{11}{5}, \frac{18}{5} \right)$$

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3.2 Equation of Straight Line in Various Standard Forms

A straight line in the coordinate plane can be represented by various forms. Each form is useful for specific geometric or algebraic purposes.

3.2.1 Slope-Intercept Form:

The slope-intercept form of the equation of a line is:

$$y = mx + c$$

Where:

• $m$ is the slope (gradient) of the line, which represents the rate of change of $y$ with respect to $x$.
• $c$ is the y-intercept, the point where the line intersects the $y$-axis.

Example: If a line has a slope of $2$ and a y-intercept of $-3$, its equation is:

$$y = 2x - 3$$

3.2.2 Intercept Form:

The intercept form of a line is:

$$\frac{x}{a} + \frac{y}{b} = 1$$

Where:

• $a$ is the x-intercept, the point where the line intersects the $x$-axis.
• $b$ is the y-intercept, the point where the line intersects the $y$-axis.

Example: If the line intersects the $x$-axis at $4$ and the $y$-axis at $6$, the equation is:

$$\frac{x}{4} + \frac{y}{6} = 1$$

3.2.3 Perpendicular Form:

The perpendicular form of the equation of a line is:

$$x \cos \theta + y \sin \theta = p$$

Where:

• $\theta$ is the angle made by the line with the positive $x$-axis.
• $p$ is the perpendicular distance from the origin to the line.

Example: For a line making an angle of $30^\circ$ with the positive $x$-axis and having a perpendicular distance of $5$ from the origin, the equation is:

$$x \cos 30^\circ + y \sin 30^\circ = 5$$

Using values for cosine and sine of $30^\circ$:

$$x \cdot \frac{\sqrt{3}}{2} + y \cdot \frac{1}{2} = 5$$

3.2.4 One-Point Slope Form:

The one-point slope form of a line is:

$$y - y_1 = m(x - x_1)$$

Where:

• $(x_1,y_1)\; is\; a\; known\; point\; on\; the\; line.$
• $m$ is the slope of the line.

Example: For a line passing through $P(3, 4)$ with slope $2$, the equation is:

$$y - 4 = 2(x - 3)$$

3.2.5 Two-Point Form:

The two-point form of a line is:

$$\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}$$

Where:

• $(x_1,y_1)\; and$$(x_2,y_2)\; are\; two\; known\; points\; on\; the\; line.$

Example: For points $A(1, 2)$ and $B(3,6),\; the\; equation\; is:$

$$\frac{y - 2}{6 - 2} = \frac{x - 1}{3 - 1}$$ $$\frac{y - 2}{4} = \frac{x - 1}{2}$$

3.2.6 General Form:

The general form of the equation of a straight line is:

$$Ax + By + C = 0$$

Where $A$, $B$, and $C$ are constants.

Example: For a line with $A = 2$, $B = -3$, and $C = 5$, the equation is:

$$2x - 3y + 5 = 0$$

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3.3 Angle Between Two Lines, Parallel and Perpendicular Lines

Angle Between Two Lines:

The angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by the formula:

$$\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$$

Example: For lines with slopes $m_1 = 2$ and $m_2 = -1$, the angle between them is:

$$\tan \theta = \left| \frac{2 - (-1)}{1 + 2(-1)} \right| = \left| \frac{3}{-1} \right| = 3$$

Thus, $\theta = \tan^{-1}(3)$.

Parallel Lines:

Two lines are parallel if their slopes are equal. That is:

$$m_1 = m_2$$

Perpendicular Lines:

Two lines are perpendicular if the product of their slopes is $-1$. That is:

$$m_1 \cdot m_2 = -1$$

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3.4 Perpendicular Distance Formula

The perpendicular distance $d$ from a point $(x_1,y_1)\; to\; a\; line$$Ax+By+C=\mathrm{0\; is\; given\; by\; the\; formula:}$

$$d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$$

Example: Find the perpendicular distance from the point $P(3, 4)$ to the line $2x - 3y + 5 = 0$.

Using the formula:

$$d = \frac{|2(3) - 3(4) + 5|}{\sqrt{2^2 + (-3)^2}} = \frac{|6 - 12 + 5|}{\sqrt{4 + 9}} = \frac{| -1 |}{\sqrt{13}} = \frac{1}{\sqrt{13}}$$

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Summary of Key Formulas:

1. Distance Formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
2. Section Formula: $$P = \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right)$$
   
3. Equation of Line Forms:
   • Slope-Intercept: $y=mx+c$
   • Intercept Form: $\frac{x}{a} + \frac{y}{b} = 1$
     
   • Perpendicular Form: $x \cos \theta + y \sin \theta = p$
     
   • One-Point Slope: $y - y_1 = m(x - x_1)$
     
   • Two-Point Form: $\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}$
   • General Form: $Ax + By + C = 0$
     
4. Angle Between Two Lines: $$\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$$
5. Perpendicular Distance Formula: $$d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$$​

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