Coordinate Geometry

Introduction

Coordinate Geometry, also known as Analytic Geometry, is a branch of mathematics that uses algebraic equations to describe geometric properties and relationships. It plays a crucial role in the JEE Main Mathematics syllabus, providing a foundation for understanding various geometric concepts through algebraic methods. This study note will cover essential topics, breaking down complex ideas into digestible sections and providing examples to enhance understanding.

1. Coordinate System

1.1 Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional plane defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is called the origin, denoted by $(0,0)$.

• Coordinates of a Point: Any point in the plane can be represented by an ordered pair $(x, y)$, where $x$ is the horizontal distance from the origin and $y$ is the vertical distance.

Example

Example: The point $(3, 4)$ lies 3 units to the right of the origin and 4 units above the origin.

1.2 Distance Formula

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in the Cartesian plane is given by: $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Example

Example: The distance between the points $(1, 2)$ and $(4, 6)$ is: $$ d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{9 + 16} = 5 $$

1.3 Section Formula

The section formula determines the coordinates of a point dividing a line segment joining two points $(x_1, y_1)$ and $(x_2, y_2)$ in the ratio $m:n$.

• Internal Division: $$ (x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) $$
• External Division: $$ (x, y) = \left( \frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n} \right) $$

Tip

For internal division, remember to use the sum of the ratios ($m+n$) in the denominator. For external division, use the difference ($m-n$).

2. Straight Lines

2.1 Slope of a Line

The slope (or gradient) of a line is a measure of its steepness and is defined as the ratio of the vertical change to the horizontal change between two points on the line.

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

2.2 Equation of a Line

Several forms of the equation of a line are used in coordinate geometry:

• Slope-Intercept Form: $$ y = mx + c $$ where $m$ is the slope and $c$ is the y-intercept.
• Point-Slope Form: $$ y - y_1 = m(x - x_1) $$ where $(x_1, y_1)$ is a point on the line.
• Two-Point Form: $$ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) $$
• General Form: $$ Ax + By + C = 0 $$

Example

Example: Find the equation of a line passing through the points $(1, 2)$ and $(3, 4)$.

Using the two-point form: $$ y - 2 = \frac{4 - 2}{3 - 1}(x - 1) \Rightarrow y - 2 = x - 1 \Rightarrow y = x + 1 $$

2.3 Angle Between Two Lines

The angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by: $$ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| $$

Note

If the lines are perpendicular, $m_1 m_2 = -1$. If they are parallel, $m_1 = m_2$.

3. Circles

3.1 Equation of a Circle

The standard form of the equation of a circle with center $(h, k)$ and radius $r$ is: $$ (x - h)^2 + (y - k)^2 = r^2 $$

3.2 General Form

The general form of a circle's equation is: $$ x^2 + y^2 + 2gx + 2fy + c = 0 $$ where the center is $(-g, -f)$ and the radius is $\sqrt{g^2 + f^2 - c}$.

Example

Example: Find the center and radius of the circle given by the equation $x^2 + y^2 - 4x + 6y - 12 = 0$.

Rewriting in standard form: $$ (x - 2)^2 + (y + 3)^2 = 25 $$ The center is $(2, -3)$ and the radius is $5$.

4. Parabolas

4.1 Standard Forms

A parabola is a set of points equidistant from a fixed point (focus) and a fixed line (directrix).

• Vertical Parabola: $(x - h)^2 = 4a(y - k)$
• Horizontal Parabola: $(y - k)^2 = 4a(x - h)$

4.2 Focus and Directrix

For the parabola $y^2 = 4ax$:

• Focus: $(a, 0)$
• Directrix: $x = -a$

Example

Example: Find the focus and directrix of the parabola $y^2 = 12x$.

Here, $4a = 12 \Rightarrow a = 3$. Thus:

• Focus: $(3, 0)$

• Directrix: $x = -3$

5. Ellipses

5.1 Standard Form

An ellipse is the set of points where the sum of distances from two fixed points (foci) is constant.

• Horizontal Ellipse: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$
• Vertical Ellipse: $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$

5.2 Foci and Eccentricity

For the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$:

• Foci: $(\pm c, 0)$ where $c = \sqrt{a^2 - b^2}$
• Eccentricity: $e = \frac{c}{a}$

Example

Example: Find the foci and eccentricity of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$.

Here, $a^2 = 16$ and $b^2 = 9 \Rightarrow c = \sqrt{16 - 9} = \sqrt{7}$. Thus:

• Foci: $(\pm \sqrt{7}, 0)$

• Eccentricity: $e = \frac{\sqrt{7}}{4}$

6. Hyperbolas

6.1 Standard Form

A hyperbola is the set of points where the difference of distances from two fixed points (foci) is constant.

• Horizontal Hyperbola: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$
• Vertical Hyperbola: $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$

6.2 Foci and Asymptotes

For the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$:

• Foci: $(\pm c, 0)$ where $c = \sqrt{a^2 + b^2}$
• Asymptotes: $y = \pm \frac{b}{a}x$

Example

Example: Find the foci and asymptotes of the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$.

Here, $a^2 = 9$ and $b^2 = 16 \Rightarrow c = \sqrt{9 + 16} = 5$. Thus:

• Foci: $(\pm 5, 0)$

• Asymptotes: $y = \pm \frac{4}{3}x$

Conclusion

Coordinate Geometry is a powerful tool that bridges algebra and geometry, allowing for the analysis and solution of complex geometric problems using algebraic methods. Mastery of the concepts in this study note will provide a solid foundation for tackling related problems in the JEE Main Mathematics syllabus.