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Straight lines is usually the first chapter of coordinate geometry, and it looks easy because every formula is short. Marks still go missing: a sign dropped in the intercept form, a distance between parallel lines computed without matching coefficients, an obtuse angle given when the question wanted the acute one. This post sets out the forms you need, works through distances and angles with checked numbers, and ends with a routine that builds speed.
What does the syllabus cover?
The JEE (Main) 2026 syllabus lists, under co-ordinate geometry, slope, parallel and perpendicular lines, "various forms of equations of a line, intersection of lines, angles between two lines, conditions for concurrence of three lines, the distance of a point form a line", and the centroid, orthocentre and circumcentre of a triangle. The JEE (Advanced) 2026 brochure adds lines through the intersection of two lines and the bisector of the angle between two lines.
For school, NCERT's Class 11 chapter on straight lines covers slope, the main forms and the distance of a point from a line. The CBSE 2026-27 mathematics curriculum says the normal form and the general equation of a line will be assessed only formatively in Class 11. Check the documents for your own exam year.
Which form of the equation should you use?
Pick the form that matches what the question gives you:
Point-slope: y − y₁ = m(x − x₁), when you know one point and the slope.
Two-point: slope m = (y₂ − y₁)/(x₂ − x₁), then point-slope.
Slope-intercept: y = mx + c, when you know the slope and the y-intercept.
Intercept form: x/a + y/b = 1, when you know where the line cuts both axes.
General form: Ax + By + C = 0, with slope −A/B. Convert to this form before using any distance formula.
Example. Find the line through A(1, 3) and B(4, 9), and its intercepts.
Slope m = (9 − 3)/(4 − 1) = 6/3 = 2.
Point-slope: y − 3 = 2(x − 1), so y = 2x + 1, or 2x − y + 1 = 0.
Intercepts: put y = 0 to get x = −1/2; put x = 0 to get y = 1. In intercept form, x/(−1/2) + y/1 = 1.
Check: x/(−1/2) + y = 1 means −2x + y = 1, which is the same line.
How do you find the distance from a point to a line?
The distance from (x₀, y₀) to Ax + By + C = 0 is |Ax₀ + By₀ + C| / √(A² + B²).
Example. Distance from P(3, −1) to 3x + 4y − 10 = 0.
|3(3) + 4(−1) − 10| / √(9 + 16) = |9 − 4 − 10| / 5 = 5/5 = 1.
Now do it a second way, by finding the foot of the perpendicular. The foot is (x₀ + At, y₀ + Bt) with t = −(Ax₀ + By₀ + C)/(A² + B²) = −(−5)/25 = 1/5. That gives (3 + 3/5, −1 + 4/5) = (18/5, −1/5). Check it lies on the line: 3(18/5) + 4(−1/5) = 54/5 − 4/5 = 10. The distance from P is √((3/5)² + (4/5)²) = √(25/25) = 1. Both methods agree.
Solving one problem two ways is not wasted time. In an experiment with 70 seventh-grade students learning to solve equations, Rittle-Johnson and Star (2007) found that students who compared alternative solution methods side by side made greater gains in procedural knowledge and flexibility than students who studied the same methods one at a time, with similar gains in conceptual knowledge.
What about the distance between parallel lines?
Example. Find the distance between 3x + 4y − 10 = 0 and 6x + 8y + 5 = 0.
Divide the second equation by 2: 3x + 4y + 2.5 = 0. Now A and B match, so the distance is |−10 − 2.5|/5 = 12.5/5 = 2.5. A common wrong answer is |−10 − 5|/5 = 3, from skipping the scaling step.
How do you find the angle between two lines?
For slopes m₁ and m₂, tan θ = |(m₂ − m₁)/(1 + m₁m₂)| gives the acute angle.
Example. The lines y = 2x + 1 and y = −3x + 4 have m₁ = 2 and m₂ = −3. Then (−3 − 2)/(1 + (2)(−3)) = (−5)/(−5) = 1, so θ = 45°. The obtuse angle between them is 135°.
Two special cases: the lines are parallel when m₁ = m₂, and perpendicular when m₁m₂ = −1, which makes the denominator zero. A vertical line has no slope, so handle it separately.
When are three lines concurrent?
Solve two of them and put the point into the third.
Example. For which k do x + y = 5, 2x − y = 1 and kx + 2y = 10 pass through one point? Adding the first two gives 3x = 6, so x = 2 and y = 3. Then 2k + 6 = 10, so k = 2.
Which mistakes cost the most marks?
Slope from the general form with the wrong sign. For Ax + By + C = 0 the slope is −A/B, not A/B.
Dropping the absolute value in the distance formula and reporting a negative distance.
Parallel lines with unmatched coefficients, as in the example above.
Intercept form with a negative intercept. Keep the sign in the denominator: x/(−1/2), not x/(1/2).
Ignoring vertical lines when a question says "a line through (2, 3)". The line x = 2 is a valid answer to check.
A practice routine
Forms drill (twice a week): ten lines given in mixed ways (two points, intercepts, slope and point). Write each in general form within a minute.
Two-method problems (weekly): solve three distance or foot-of-perpendicular problems by formula and by a second route, and compare.
Mixed set (weekly): eight problems mixing lines with circles in coordinate geometry and other chapters, so you choose the method yourself.
Sketch first. A rough graph, by hand or with a graphing tool, tells you whether an answer of 2.5 or 25 is plausible.
When your answer and the key disagree and you cannot find the slip, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning, so you can compare its first step with yours.
Today, take three distance problems from your last test and solve each one a second way.
