Chapter 10 – Coordinate Geometry (JEE)
1. Introduction to Coordinate Geometry
Coordinate Geometry combines **algebra and geometry** by representing points, lines,
and curves using numbers and equations.
2. Cartesian Coordinate System
A point in a plane is represented as $(x, y)$ where:
- $x$ is the x-coordinate (abscissa)
- $y$ is the y-coordinate (ordinate)
Quadrants:
- 1st: $(+,+)$
- 2nd: $(-,+)$
- 3rd: $(-,-)$
- 4th: $(+,-)$
3. Distance Formula
Distance between $(x_1,y_1)$ and $(x_2,y_2)$:
$$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
4. Section Formula
If a point divides a line joining $A(x_1,y_1)$ and $B(x_2,y_2)$ internally in ratio $m:n$:
$$\left(\frac{mx_2+nx_1}{m+n},\;\frac{my_2+ny_1}{m+n}\right)$$
External division has a similar formula with sign change.
5. Area of Triangle
Area =
$$\frac12\left|
\begin{vmatrix}
x_1 & y_1 & 1\\
x_2 & y_2 & 1\\
x_3 & y_3 & 1
\end{vmatrix}
\right|$$
If area = 0 → points are collinear.
6. Locus of a Point
The locus of a point is the **path traced** by it while satisfying a given condition.
Most locus problems reduce to distance equality.
7. Straight Line – Basic Concepts
General equation of a line:
$$ax + by + c = 0$$
Slope of the line:
$$m = -\frac{a}{b}$$
8. Slope of a Line
Slope between two points:
$$m = \frac{y_2-y_1}{x_2-x_1}$$
9. Equation of Line in Different Forms
| Form | Equation |
|---|---|
| Slope-intercept | $y = mx + c$ |
| Point-slope | $y-y_1 = m(x-x_1)$ |
| Intercept | $\frac{x}{a} + \frac{y}{b} = 1$ |
| Normal form | $x\cos\alpha + y\sin\alpha = p$ |
10. Angle Between Two Lines
$$\tan\theta = \left|\frac{m_1-m_2}{1+m_1m_2}\right|$$
If $m_1m_2=-1$ → lines are perpendicular.
11. Distance of a Point from a Line
Distance from $(x_1,y_1)$ to $ax+by+c=0$:
$$\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}$$
12. Circle – Standard Equation
$$(x-h)^2+(y-k)^2=r^2$$
Center $(h,k)$, radius $r$
13. General Equation of Circle
$$x^2+y^2+2gx+2fy+c=0$$
Center $(-g,-f)$, radius $\sqrt{g^2+f^2-c}$
14. Parabola
Standard equation:
$$y^2=4ax$$
Focus $(a,0)$, Directrix $x=-a$
15. Ellipse
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
$a>b$, foci at $(\pm ae,0)$
16. Hyperbola
$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$
Asymptotes: $y=\pm\frac{b}{a}x$
17. Pair of Straight Lines
General second degree equation:
$$ax^2+2hxy+by^2=0$$
Used heavily in JEE Advanced.
18. Common JEE Traps
- Wrong slope sign
- Forgetting modulus in distance
- Confusing center and focus
- Ignoring condition for tangency
19. Typical JEE Question Patterns
| Topic | Focus |
|---|---|
| Straight lines | Slope, distance, angle |
| Circle | Center, radius, tangent |
| Conics | Focus-directrix property |
20. Final Revision Checklist
You have mastered Coordinate Geometry if you can:
- Visualize graphs clearly
- Switch between forms of equations
- Handle locus problems confidently
- Solve line-circle-conic problems accurately