Formula Sheet — Straight Lines

Card 1: Slope

Fact Statement
Slope from inclination m=tan⁡θm = \tan\theta, θ≠90°\theta \ne 90°
Slope from two points m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}, x1≠x2x_1 \ne x_2
Horizontal / vertical slope 00 / slope undefined
Parallel m1=m2m_1 = m_2
Perpendicular m1m2=−1m_1 m_2 = -1
Angle between lines tan⁡θ=∣m2−m11+m1m2∣\tan\theta = \left\lvert\frac{m_2 - m_1}{1 + m_1m_2}\right\rvert, 1+m1m2≠01 + m_1m_2 \ne 0
Collinearity of A,B,CA, B, C slope of ABAB = slope of BCBC

Card 2: Equation forms

Form Equation
Horizontal / vertical y=±ay = \pm a / x=±bx = \pm b
Point-slope y−y0=m(x−x0)y - y_0 = m(x - x_0)
Two-point y−y1=y2−y1x2−x1(x−x1)y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)
Slope-intercept y=mx+cy = mx + c; with xx-intercept dd: y=m(x−d)y = m(x - d)
Intercept xa+yb=1\frac{x}{a} + \frac{y}{b} = 1
General Ax+By+C=0Ax + By + C = 0, slope −AB-\frac{A}{B} (B≠0B \ne 0)

Card 3: Distances

Quantity Formula
Point to line d=∣Ax1+By1+C∣A2+B2d = \frac{\lvert Ax_1 + By_1 + C\rvert}{\sqrt{A^2 + B^2}}
Origin to line d=∣C∣A2+B2d = \frac{\lvert C\rvert}{\sqrt{A^2 + B^2}}
Between parallels Ax+By+C1=0Ax + By + C_1 = 0, Ax+By+C2=0Ax + By + C_2 = 0 d=∣C1−C2∣A2+B2d = \frac{\lvert C_1 - C_2\rvert}{\sqrt{A^2 + B^2}}

Card 4: JEE Corner machinery

Tool Statement
Concurrency of three lines intersection of two lies on the third; determinant of coefficients =0= 0
Family through intersection L1+λL2=0L_1 + \lambda L_2 = 0
Foot of perpendicular (x1−At, y1−Bt)(x_1 - At,\ y_1 - Bt) with t=Ax1+By1+CA2+B2t = \frac{Ax_1 + By_1 + C}{A^2 + B^2}
Image in the line (x1−2At, y1−2Bt)(x_1 - 2At,\ y_1 - 2Bt), same tt
Point at distance rr along direction θ\theta (x1+rcos⁡θ, y1+rsin⁡θ)(x_1 + r\cos\theta,\ y_1 + r\sin\theta)
Area from y=m1x+c1y = m_1x + c_1, y=m2x+c2y = m_2x + c_2, x=0x = 0 (c1−c2)22∣m1−m2∣\frac{(c_1 - c_2)^2}{2\lvert m_1 - m_2\rvert}
Parallel / perpendicular through (x1,y1)(x_1, y_1) A(x−x1)+B(y−y1)=0A(x - x_1) + B(y - y_1) = 0 / B(x−x1)−A(y−y1)=0B(x - x_1) - A(y - y_1) = 0

Last-Minute Mistake Checklist

  1. Vertical lines have no slope — never write m=tan⁡90°m = \tan 90°; handle x=bx = b separately in every slope argument.
  2. The angle formula needs the absolute value for the acute angle; dropping it silently returns the obtuse companion, and the pair always sums to 180°180°.
  3. Perpendicularity is m1m2=−1m_1m_2 = -1, not m1=−m2m_1 = -m_2 — the second is a reflection condition, not perpendicularity.
  4. Convert to general form before any distance computation — substituting into 12(x+6)=5(y−2)12(x + 6) = 5(y - 2) directly gives a wrong (but plausible-looking) number.
  5. Match coefficients before using ∣C1−C2∣A2+B2\frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} — for 3x−4y+7=03x - 4y + 7 = 0 and 6x−8y+5=06x - 8y + 5 = 0, rescale first (d=910d = \frac{9}{10}, not 25\frac{2}{5}).
  6. Intercepts can be negative — in xa+yb=1\frac{x}{a} + \frac{y}{b} = 1, keep the sign of aa and bb; "equal intercepts" through a point can force a negative common value.
  7. In the foot/image formula, remember the denominator A2+B2A^2 + B^2 in tt, and the factor 2 for the image; sanity-check that the midpoint of point and image lies on the line.
  8. "Distance along a line" is not perpendicular distance — parametrise as (x1+rcos⁡θ,y1+rsin⁡θ)(x_1 + r\cos\theta, y_1 + r\sin\theta) and solve for rr; it is always ≥\ge the perpendicular distance.
  9. Family answer check: after finding λ\lambda in L1+λL2=0L_1 + \lambda L_2 = 0, substitute the intersection point back — it must satisfy your final line exactly.
  10. Present the equation in the requested form (intercept, general, slope-intercept) with integer coefficients where possible — marks are tied to the stated form.

How this chapter flows into the next: Conic Sections puts circles, parabolas, ellipses and hyperbolas on this same coordinate plane. Chords, tangents and axes of symmetry there are all straight lines — every formula on these cards returns immediately.