JEE Mathematics Straight Lines

Distance Formula:
\( d = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2} \).

Section Formula :
\( x = \frac{mx_2 \pm nx_1}{m \pm n} ; y = \frac{my_2 \pm ny_1}{m \pm n} \).

Centroid, Incentre & Excentre:
Centroid \( G \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \), Incentre \( I \left( \frac{ax_1 + bx_2 + cx_3}{a+b+c}, \frac{ay_1 + by_2 + cy_3}{a+b+c} \right) \)
Excentre \( I_1 \left( \frac{-ax_1 + bx_2 + cx_3}{-a+b+c}, \frac{-ay_1 + by_2 + cy_3}{-a+b+c} \right) \)

Area of a Triangle:
\( \Delta ABC = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \)

Slope Formula:
Line Joining two points \( (x_1, y_1) \) & \( (x_2, y_2) \), \( m = \frac{y_1 - y_2}{x_1 - x_2} \)

Condition of collinearity of three points:
\( \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = 0 \)

Angle between two straight lines:
\( \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \)

Two Lines: \( ax + by + c = 0 \) and \( a'x + b'y + c' = 0 \) two lines
1. parallel if \( \frac{a}{a'} = \frac{b}{b'} \neq \frac{c}{c'} \).
2. Distance between two parallel lines = \( \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}} \).
3 Perpendicular : If \( aa' + bb' = 0 \).

A point and line:
1. Distance between point and line = \( \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \).
2. Reflection of a point about a line: \( \frac{x - x_1}{a} = \frac{y - y_1}{b} = -2 \frac{ax_1 + by_1 + c}{a^2 + b^2} \)
3. Foot of the perpendicular from a point on the line is \( \frac{x - x_1}{a} = \frac{y - y_1}{b} = - \frac{ax_1 + by_1 + c}{a^2 + b^2} \)

Bisectors of the angles between two lines:
\( \frac{ax + by + c}{\sqrt{a^2 + b^2}} = \pm \frac{a'x + b'y + c'}{\sqrt{a'^2 + b'^2}} \)

Condition of Concurrency of three straight lines \( a_ix + b_iy + c_i = 0, i = 1,2,3 \) is
\( \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = 0 \).

A Pair of straight lines through origin: \( ax^2 + 2hxy + by^2 = 0 \)
If \( \theta \) is the acute angle between the pair of straight lines, then \( \tan \theta = \frac{2\sqrt{h^2 - ab}}{|a + b|} \).