Straight Lines and Pair of Straight Lines
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Direct answer
A line is fixed by a point and slope: through (x1, y1) with slope m it is y - y1 = m(x - x1), and in general form ax + by + c = 0 its slope is -a/b. Nearly every JEE question is distance, angle or family logic — the perpendicular distance from (x1, y1) is |ax1 + by1 + c|/sqrt(a^2 + b^2), and the angle between slopes m1, m2 has tan theta = |(m1 - m2)/(1 + m1 m2)|.
What you must remember
- Slope m = tan theta = (y2 - y1)/(x2 - x1); parallels share slope, perpendiculars satisfy m1 m2 = -1, verticals have undefined slope.
- Standard forms: point-slope; two-point; y = mx + c; x/a + y/b = 1; normal x cos alpha + y sin alpha = p.
- Distances: point to line |ax1 + by1 + c|/sqrt(a^2 + b^2); between parallels |c1 - c2|/sqrt(a^2 + b^2) after matching the x and y coefficients.
- Foot and image: for ax + by + c = 0 and point (x1, y1), with t = (ax1 + by1 + c)/(a^2 + b^2), foot (x1 - at, y1 - bt), image (x1 - 2at, y1 - 2bt).
- Angle bisectors: (a1x + b1y + c1)/sqrt(a1^2 + b1^2) = ± (a2x + b2y + c2)/sqrt(a2^2 + b2^2); pick the sign by testing a point for the acute bisector.
- Concurrency: the determinant of the three lines' coefficients vanishes; the family through the intersection of L1 = 0 and L2 = 0 is L1 + lambda L2 = 0.
- Pair of lines through the origin: ax^2 + 2hxy + by^2 = 0 gives two lines when h^2 >= ab, with slopes summing to -2h/b and multiplying to a/b; the angle between them has tan theta = |2 sqrt(h^2 - ab)/(a + b)|; the general second-degree equation is a pair of lines when abc + 2fgh - af^2 - bg^2 - ch^2 = 0.
Common confusion
The slope machinery silently excludes verticals — a vertical and a horizontal line are perpendicular though neither satisfies m1 m2 = -1. In pair-of-lines work the 2h convention is the standing trip: reading h where the equation offers 2h flips every formula downstream. And |c1 - c2| for parallels is valid only after matching the x and y coefficients.
Exam-focused takeaway
JEE Main asks distances, angles, foot and image, concurrency and pair-of-lines angles as quick numericals. JEE Advanced prefers geometric constructions: shortest paths via reflection, loci from line constraints, bisectors placing a point equidistant from two lines, and joint equations of the lines from the origin to a conic-line intersection. Draw the configuration first — these questions are lost in the sketch more often than in the algebra.
Frequently asked questions
What is the distance between two parallel lines?
After matching the x and y coefficients: |c1 - c2|/sqrt(a^2 + b^2), else invalid.
How do I find the image of a point in a line?
From the foot (x1 - at, y1 - bt), extend equally: the image is (x1 - 2at, y1 - 2bt) with t = (ax1 + by1 + c)/(a^2 + b^2).
When are three given lines concurrent?
When the determinant of their coefficients is zero; otherwise solve two and test the third.
What does h^2 = ab mean for a pair of straight lines?
The two lines coincide into one; h^2 > ab gives distinct real lines, h^2 < ab no real pair.
How do I write the joint equation of the lines joining the origin to a curve's intersection with a line?
Homogenise the curve's equation with the line's linear form to equal degree — a standard Advanced move.