Fundamentals
The core facts every aspirant should own — each a titled nugget with a real-world story, the concept in plain words, and a memory trick. Works even when the internet doesn't.
300 fundamentals
“If you only know a line’s slope and one point, using the two-point formula is like trying to draw a river with just one bank.”
When only the slope and a single point are given, picking the two-point form forces you to invent a second point, leading to wrong equations. Use point-slope form directly; it avoids unnecessary calculations and reduces sign mistakes.
Memory trick: Match given data → pick matching form.
“Surveyors determine a hill’s steepness by measuring two altitude posts; the rise over run gives the slope m = (y2−y1)/(x2−x1).”
The slope between (x1,y1) and (x2,y2) is m = (y2−y1)/(x2−x1). It measures the line’s steepness. If x1 = x2 the line is vertical and the slope is undefined; JEE may ask to state ‘no slope’ in that case.
m = (y₂−y₁)/(x₂−x₁)
Memory trick: Δy/Δx, only if Δx ≠ 0.
“The shortest walk from a lamp post to a straight road is a perpendicular; its length is d = |ax0+by0+c|/√(a²+b²).”
For a line ax+by+c=0 and a point (x0,y0), the perpendicular distance is d = |ax0+by0+c|/√(a²+b²). The numerator is the line’s left side evaluated at the point, absolute value ensures positivity. Forgetting the absolute value flips sign errors.
d = |ax₀+by₀+c|/√(a²+b²)
Memory trick: d = |ax0+by0+c| / √(a²+b²).
“Two railway tracks that never meet are parallel, while a ladder against a wall is perpendicular, their slopes obey m1=m2 or m1·m2=−1 (0·∞ case).”
Parallel lines share the same slope; vertical lines are parallel even though their slope is undefined. Perpendicular lines obey m1·m2 = –1; for a horizontal (m=0) and vertical (undefined) pair the right-angle condition holds despite the product not being definable. Exam trap: don’t force the –1 product on a 0–∞ case.
Memory trick: Parallel = same m; Perp = m1·m2 = –1 (or 0 & ∞)
“Writing y=mx+c is like setting a car’s speed (m) and its starting mileage (c) before a trip.”
Use y=mx+c when the slope m and y-intercept c are known; use (y−y1)=m(x−x1) when a point (x1,y1) and the slope are given. Both equations describe the same line; avoid substituting a point into y=mx+c unless you first determine c.
Memory trick: Intercept first → y=mx+c; point first → (y−y1)=m(x−x1).
“Two ladders leaning against a wall at the same angle have identical slopes, so they never cross.”
Equal slopes (m1=m2) make lines parallel, not perpendicular. For perpendicular lines the slopes multiply to –1 (or one line is vertical and the other horizontal). The myth confuses “right angle” with “same steepness.”
Memory trick: Parallel = Same; Perp = Product –1.
“Eccentricity e tells how stretched a conic is, like tightening a rubber band from a circle to a parabola and then to a hyperbola.”
Eccentricity e = distance focus-centre / semi-major axis. Values: circle e=0, ellipse 0<e<1, parabola e=1, hyperbola e>1. It is dimensionless and same for all similar conics. Do not assume only hyperbola has e>1; ellipse also has a positive e less than 1.
Memory trick: e: 0-circle, 1-parabola, >1-hyperbola
“A basketball (circle, e=0), an Olympic track (ellipse, e<1), a water-spray arc (parabola, e=1) and a cooling-tower silhouette (hyperbola, e>1).”
Eccentricity e classifies conics: e=0 → circle, 0<e<1 → ellipse, e=1 → parabola, e>1 → hyperbola. It is the constant ratio PF/PD (focus to directrix). Trap: forgetting a circle is a special ellipse (e=0) and that hyperbola always has two separate branches.
Memory trick: e: 0-circle, <1-ellipse, =1-parabola, >1-hyperbola.
“A flashlight’s reflector cross-section is a parabola: the bulb sits at the focus and a straight line behind it is the directrix.”
A conic consists of points P such that PF = e·PD, where F is a fixed point (focus) and D is a fixed line (directrix). The eccentricity e determines the type: e<1 ellipse, e=1 parabola, e>1 hyperbola. Remember that the ratio uses absolute distances; a common trap is to mix up the sign of PD when the point lies on the opposite side of the directrix.
Memory trick: F = point, D = line – point vs line.
“In a satellite dish, the receiver at the focus collects reflected signals; the straight line behind the dish is the directrix.”
A conic is the set of points P with PF/PD = e, where F is a fixed point (focus) and D a fixed line (directrix). For a parabola e=1, PF = PD; for an ellipse e<1, for a hyperbola e>1. The common trap is treating the curve itself as the directrix or swapping point and line, which gives wrong equations.
Memory trick: Focus = point (F), Directrix = line (D) → remember F-point, D-line
“The equation (x−h)²+(y−k)²=r² is the GPS map for any circle, just like latitude-longitude pins its centre.”
The circle’s standard form (x−h)²+(y−k)²=r² gives centre (h,k) and radius r. The parabola’s basic forms are y²=4ax (opens right) and x²=4ay (opens up). Mixing the variables flips the opening direction, a frequent slip in JEE.
Memory trick: y²=4ax opens right; x²=4ay opens up.
“If you write y²=4ax but draw the curve opening upward, you’ve swapped the axis – the equation tells the direction.”
When the squared term is y², the parabola opens horizontally (right for positive a); when it is x², it opens vertically (up for positive a). Using the wrong axis produces an incorrect graph and wrong focus/directrix.
Memory trick: Square y → opens right; square x → opens up.
“A round pothole on the road is described by (x−h)²+(y−k)²=r², where (h,k) is the centre you’d mark on a map.”
For any circle, (x−h)²+(y−k)²=r² encodes its centre (h,k) and radius r. It works for circles anywhere on the plane. A common error is writing (x+h)² or (y+k)², which shifts the centre opposite to intended.
(x−h)²+(y−k)² = r²
Memory trick: Center (h,k), radius r: (x−h)²+(y−k)²=r².
“A satellite dish reflects signals toward its focus, just like y² = 4ax curves rightward toward (a,0).”
The curve y² = 4ax is a parabola with vertex at (0,0), focus at (a,0) and directrix x = –a. Any point is equidistant from focus and directrix. In exams, watch the sign of a: a>0 opens right, a<0 opens left; also ensure axis is the x-axis.
y² = 4ax
Memory trick: Focus (a,0), directrix x=−a → opens right if a>0
“A planet’s closed orbit around the Sun is an ellipse, while a comet that escapes follows an open hyperbola.”
An ellipse is a closed curve with eccentricity e < 1; the sum of distances from any point to the two foci is constant (2a). A hyperbola is an open two-branch curve with e > 1; the absolute difference of distances to the foci is constant (2a). Remember that e = 1 marks a parabola, not a hyperbola.
Memory trick: EllipSe = Sum constant; Hyperbola = Difference constant.
“In a Cassegrain telescope the primary mirror is a parabola and the secondary hyperbola redirects light from one focus to the other.”
A parabola has eccentricity exactly 1, one focus and a directrix; its points satisfy equal distance to focus and directrix. A hyperbola has e > 1, two foci, and the absolute difference of distances to them is constant. Beware mixing up e > 1 with the parabola’s e = 1 when identifying the curve.
Memory trick: Parabola = 1 focus, e=1; Hyperbola = 2 foci, e>1.
“A satellite’s elliptical orbit keeps the ratio of its distance to Earth (focus) and to the straight line (directrix) fixed – that ratio is eccentricity.”
For any point P on a conic, e = PF ÷ PD, where PF is the distance to the focus and PD the perpendicular distance to the directrix. This ratio is constant for the whole curve. The mistake is treating e as the focus-directrix distance or a variable; it is a fixed ratio, giving e=0 (circle), 0<e<1 (ellipse), e=1 (parabola), e>1 (hyperbola).
Memory trick: E = PF/PD – ‘PF over PD’ sounds like a photo (P-F-O-P-D).
“A door knob may be missing, but you can still feel the direction the door wants to turn – the limit exists without the function value.”
A limit can exist even if the function is not defined at that point; it cares only about nearby values. The common mistake is to equate limit with f(a). Always verify existence of the approaching value first.
Memory trick: Limit ≠ f(a) if f(a) missing.
“When you slide a phone screen across a table, the motion feels smooth only if the path has no sudden jumps.”
A function f is continuous at x = a when limₓ→a f(x) exists and equals f(a). Hence there is no hole, jump or break at that point. The exam trap is assuming any piecewise or |x| graph is discontinuous without checking the limit; |x| is continuous at 0 because both sides approach 0.
Memory trick: Continuity: limit = value → L=V
“Zooming into a tiny slice of a pizza, the chord length and the arc become almost the same – that’s why sin x/x →1.”
The standard limit lim (x→0) sin x / x = 1 resolves the 0/0 indeterminate form; it underlies many JEE limit problems. Remember to work in radians; otherwise the limit equals π/180, not 1.
Memory trick: sin x/x →1 (rad).
“When you tilt a tiny angle, the arc length and chord become almost the same, so sin θ/θ ≈1.”
The limit limₓ→0 (sin x)/x equals 1. It follows from the squeeze theorem on the unit circle or from the series sin x = x – x³/6… . The usual exam trap is to stop at the 0/0 form and write 0 or claim the limit is undefined.
Memory trick: Tiny angle rule: sin θ ≈ θ, so sin θ/θ → 1
“A carpenter measuring a tiny angle on a circular protractor finds the arc length equals the chord length, illustrating sin θ≈θ.”
This limit holds only when x is measured in radians; in degrees sin θ/θ → π/180. The radian definition makes the arc length equal to the angle, giving the result. A common mistake is plugging degrees and getting a wrong factor.
lim(x→0) sin x / x = 1
Memory trick: Radians only: sin θ/θ→1, not in degrees
“A TV channel may show a picture that approaches a scene as you tune, yet the exact frequency may still be blank—limit ≠ value.”
The limit describes the value that f(x) approaches as x nears a point, independent of the actual function value at that point. At a removable discontinuity the limit exists but f(a) may be undefined or different. Remember to check f(a) separately; continuity is not automatic.
Memory trick: Limit ≠ f(a) unless continuous
“A river flowing without a dam represents a continuous function; a dam causing water to jump is a discontinuity.”
A continuous function has no breaks: for every a, limₓ→a f(x) = f(a). Discontinuous functions show jumps, holes (removable), or vertical asymptotes. Even if the limit exists (hole), the function is still discontinuous. Identify the type to apply the right theorem.
Memory trick: Continuous = no jumps, holes, or asymptotes