1. Concept of Locus
A Locus is defined as the set of all points that satisfy a specific geometric condition. When a point moves under this condition, the path it traces is called its locus.
Examples:
• Locus of points at a distance 'r' from a fixed point: A Circle.
• Locus of points equidistant from two fixed points: The Perpendicular Bisector.
2. Section Formula
For a point \(P(x, y)\) dividing the line segment \(A(x_1, y_1)B(x_2, y_2)\) in ratio \(m_1 : m_2\):
A. Internal Division
\[x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \quad y = \frac{m_1y_2 + m_2y_1}{m_1 + m_2}\]
B. External Division
When the point \(P\) lies outside the segment \(AB\), the formula uses subtraction:
\[x = \frac{m_1x_2 - m_2x_1}{m_1 - m_2}, \quad y = \frac{m_1y_2 - m_2y_1}{m_1 - m_2}\]
3 & 4. Slope and Slope-Intercept Form
The Slope (Gradient) \(m\) measures the inclination of a line.
\[m = \frac{y_2 - y_1}{x_2 - x_1} = \tan \theta\]
The Slope-Intercept Form:
\[y = mx + c\]
Where \(m = \text{slope}\) and \(c = \text{y-intercept}\)
5 & 6. Double-Intercept and Normal Form
Double-Intercept Form
\[\frac{x}{a} + \frac{y}{b} = 1\]
Where \(a\) = x-intercept, \(b\) = y-intercept
Normal (Perpendicular) Form
\[x \cos \alpha + y \sin \alpha = p\]
Where \(p\) = perpendicular distance from origin, \(\alpha\) = angle of the normal with positive x-axis
7. Reduction of General Equation \(Ax + By + C = 0\)
A. Reduction to Slope-Intercept Form (\(y = mx + c\))
\[y = \left(-\frac{A}{B}\right)x + \left(-\frac{C}{B}\right)\]
Slope \(m = -A/B\), y-intercept \(c = -C/B\)
B. Reduction to Point-Slope Form (\(y - y_1 = m(x - x_1)\))
\[y - y_1 = \left(-\frac{A}{B}\right)(x - x_1)\]
C. Reduction to Double-Intercept Form (\(\frac{x}{a} + \frac{y}{b} = 1\))
\[\frac{x}{(-C/A)} + \frac{y}{(-C/B)} = 1\]
x-intercept \(a = -C/A\), y-intercept \(b = -C/B\)
D. Reduction to Normal (Perpendicular) Form (\(x \cos \alpha + y \sin \alpha = p\))
\[\left(\frac{A}{\sqrt{A^2+B^2}}\right)x + \left(\frac{B}{\sqrt{A^2+B^2}}\right)y = \frac{-C}{\sqrt{A^2+B^2}}\]
Where \(\cos \alpha = \frac{A}{\sqrt{A^2+B^2}}\), \(\sin \alpha = \frac{B}{\sqrt{A^2+B^2}}\), and \(p = \frac{-C}{\sqrt{A^2+B^2}}\)