Straight Lines

Straight lines form the bedrock of coordinate geometry in JEE Mathematics. Every line in the Cartesian plane can be described by a linear equation in x and y, and mastering the various forms of this equation is essential for solving problems efficiently.

A line is uniquely determined by two conditions — two points, a point and a slope, or a point and a direction. The slope of a line passing through points (x1, y1) and (x2, y2) is m = (y2 - y1)/(x2 - x1), provided x1 is not equal to x2. The slope measures the tangent of the angle the line makes with the positive x-axis.

The principal forms of the equation of a line are: slope-intercept form y = mx + c, point-slope form y - y1 = m(x - x1), two-point form (y - y1)/(y2 - y1) = (x - x1)/(x2 - x1), intercept form x/a + y/b = 1, and the general form ax + by + c = 0. Each form is suited for different problem contexts.

The angle between two lines with slopes m1 and m2 is given by tan($\theta$) = |m1 - m2|/(1 + m1m2). Two lines are parallel when m1 = m2 and perpendicular when m1m2 = -1. The distance of a point (x1, y1) from a line ax + by + c = 0 is |ax1 + by1 + c|/$\sqrt{a^{2} + b^{2}}$. The distance between two parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is |c1 - c2|/$\sqrt{a^{2} + b^{2}}$.

A family of lines through the intersection of L₁: a1x + b1y + c1 = 0 and L₂: a2x + b2y + c2 = 0 is given by L₁ + $\lambda$*L₂ = 0. This parametric representation is extremely powerful for JEE problems where you need to find a line satisfying an additional condition.

The foot of the perpendicular from a point P(x1, y1) to a line ax + by + c = 0 and the image of P in that line are found using the formula (h - x1)/a = (k - y1)/b = -(ax1 + by1 + c)/($a^{2}$ + $b^{2}$), where for the foot we use one multiple and for the image we use twice that multiple.

Pair of straight lines through the origin is represented by $ax^{2}$ + 2hxy + $by^{2}$ = 0, where the lines are real and distinct when $h^{2}$ > ab, coincident when $h^{2}$ = ab, and imaginary when $h^{2}$ < ab.
The angle between the pair is tan($\theta$) = 2*$\sqrt{h^{2} - ab}$/|a + b|.

The general second-degree equation $ax^{2}$ + 2hxy + $by^{2}$ + 2gx + 2fy + c = 0 represents a pair of straight lines when the determinant |a h g; h b f; g f c| = 0, i.e., abc + 2fgh - $af^{2}$ - $bg^{2}$ - $ch^{2}$ = 0.

The key problem-solving concept is recognizing which form of the line equation simplifies the given constraints and using the family of lines approach to avoid solving simultaneous equations.