“Minus one means perpendicular”
Testing two lines
Parallel lines have equal slopes; perpendicular lines have slopes multiplying to −1. A vertical line is the exception, having no slope at all.
Mathematics, Class 11 and 12 · Gujarat Board, CBSE, GUJCET and JEE
Straight Lines and Conic Sections formulas
Coordinate geometry converts pictures into equations. The straight line forms are asked directly; the conic sections are asked as standard results, so the table of parabola, ellipse and hyperbola parameters is worth having in memory as one block.
20 formulas, 3 memory hooks, and the 5 mistakes that cost the most marks.
Straight lines
| What | Formula | Symbols and notes |
|---|---|---|
| Distance between two points | d = √((x₂ − x₁)² + (y₂ − y₁)²) | from Pythagoras |
| Section formula | ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)) | internal division in the ratio m:n |
| Midpoint | ((x₁ + x₂)/2, (y₁ + y₂)/2) | the section formula with m = n |
| Slope | m = (y₂ − y₁)/(x₂ − x₁) = tanθ | θ is the angle with the positive x axis |
| Slope-intercept form | y = mx + c | c is the y intercept |
| Point-slope form | y − y₁ = m(x − x₁) | the most used form in practice |
| Two-point form | (y − y₁)/(y₂ − y₁) = (x − x₁)/(x₂ − x₁) | when two points are given |
| Intercept form | x/a + y/b = 1 | a and b are the intercepts |
| Parallel and perpendicular | m₁ = m₂ and m₁m₂ = −1 | the two conditions asked most |
| Angle between lines | tanθ = |(m₁ − m₂)/(1 + m₁m₂)| | zero when parallel |
| Distance from a point to a line | d = |ax₁ + by₁ + c|/√(a² + b²) | the line as ax + by + c = 0 |
Circle
| What | Formula | Symbols and notes |
|---|---|---|
| Standard equation | (x − h)² + (y − k)² = r² | centre (h, k), radius r |
| General equation | x² + y² + 2gx + 2fy + c = 0 | centre (−g, −f), radius √(g² + f² − c) |
| Tangent condition | the distance from the centre to the line equals the radius | the standard way to test tangency |
Conic sections
| What | Formula | Symbols and notes |
|---|---|---|
| Parabola | y² = 4ax | vertex at origin, focus (a, 0), directrix x = −a, e = 1 |
| Parabola, latus rectum | length 4a | the chord through the focus perpendicular to the axis |
| Ellipse | x²/a² + y²/b² = 1, a > b | foci (±ae, 0), e = √(1 − b²/a²), e < 1 |
| Ellipse, latus rectum | 2b²/a | and the sum of focal distances is 2a |
| Hyperbola | x²/a² − y²/b² = 1 | foci (±ae, 0), e = √(1 + b²/a²), e > 1 |
| Hyperbola, asymptotes | y = ±(b/a)x | and the difference of focal distances is 2a |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“e tells the shape”
Identifying a conic
Eccentricity 0 is a circle, less than 1 an ellipse, exactly 1 a parabola, more than 1 a hyperbola. One number names the curve.
“Ellipse minus, hyperbola plus”
The two eccentricity formulas
Ellipse e = √(1 − b²/a²), hyperbola e = √(1 + b²/a²). The sign matches the sign in the curve's own equation.
Where marks go
The mistakes that cost the most.
- Using the distance formula for a point and a line. That needs the perpendicular distance formula instead.
- Forgetting that a vertical line has an undefined slope, so the perpendicular test fails there.
- Mixing up 4a in the parabola with 2a in the ellipse when finding the latus rectum.
- Taking a as the larger denominator in a hyperbola. For a hyperbola a belongs to the positive term, whichever is larger.
- Reading the centre of a circle as (g, f) instead of (−g, −f).
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