The chord of a conic that passes through a focus and is at right angles to the axis. For y² = 4ax it is 4a long.
Precisely: A line segment through a focus, perpendicular to the axis of the conic, with its ends on the conic; its length is 4a for the parabola y² = 4ax and 2b²/a for an ellipse or hyperbola.
How it works
Put the focus's x-coordinate into the equation and solve for y: the two answers are the ends of the latus rectum. For y² = 4ax, x = a gives y = 2a and −2a, so the length is 4a.
Examples
The latus rectum of y² = 12x is 12 long.
For y² = 8x the latus rectum runs from (2, 4) to (2, −4).
The ellipse x²/25 + y²/16 = 1 has a latus rectum 32/5 long.
The latus rectum helps sketch a conic, giving two more points on it.
Do not confuse: The latus rectum goes through the focus; the major axis goes through both foci and the vertices.
Used in
sketching parabolas, ellipses and hyperbolas, exam questions on conics
The more times you repeat a random experiment, the closer the experimental probability gets to the theoretical probability.
Precisely: As the number of independent trials grows, the relative frequency of an event tends to its probability.
How it works
With a few trials, luck can push the results far off; with many, the lucky runs one way and the other balance out in proportion. It says nothing about any single next trial, only about the long run.
Examples
Ten tosses might give 7 heads (0.7), but by the law of large numbers 10,000 tosses give very close to 0.5.
Rolling a die 12 times gave a 3 on 0.25 of the rolls; the law of large numbers says 600 or 6,000 rolls would come close to 1/6 ≈ 0.167.
Insurance companies rely on the law of large numbers: over many thousands of customers, the share who claim is steady.
Do not confuse: It does not mean results "even out" by the next toss: that belief is the gambler's fallacy. The balance happens in proportion, over very many trials.
Used in
experimental probability, insurance and polling, statistics and science
The smallest number that each of the given numbers divides into exactly.
Precisely: The lowest (least) common multiple of positive integers is the smallest positive integer that is a multiple of each; by prime factorisation it is the product of the greatest power of each prime factor involved.
How it works
Write each number as a product of prime powers. Take every prime that appears in any of them, each with its greatest power, and multiply. For two numbers, LCM = product ÷ HCF.
Examples
The LCM of 6 = 2 × 3 and 20 = 2² × 5 is 2² × 3 × 5 = 60.
Sonia takes 18 minutes a round and Ravi 12; they meet again at the start after the LCM of 18 and 12, 36 minutes.
The LCM of 96 and 404 is 96 × 404 ÷ 4 = 9696.
For 26 and 91, HCF × LCM = 13 × 182 = 2366 = 26 × 91.
Do not confuse: The LCM of 6, 72 and 120 is 360 and their HCF is 6, but 6 × 360 is not 6 × 72 × 120: the product rule is only for two numbers.
Used in
adding fractions (the common denominator), when repeating events line up again, bells, lights and timetables
The distance along a tangent from a point outside a circle to where the tangent touches the circle.
Precisely: For an external point P and a tangent touching the circle at T, the length PT; the two tangents from P have equal lengths, and PT² = OP² − r².
How it works
The radius OT meets the tangent at a right angle, so O, T and P make a right triangle with hypotenuse OP. Pythagoras then gives the tangent's length from the radius and the distance to the centre.
Examples
From a point 25 cm from the centre, the length of a tangent is 24 cm, so the radius is √(25² − 24²) = 7 cm.
The two tangents from one outside point have the same length of a tangent: PQ = PR.
In two concentric circles of radii 5 cm and 3 cm, a chord of the larger touching the smaller is twice a length of a tangent: 2 × √(25 − 9) = 8 cm.
Do not confuse: From a point inside the circle there is no tangent at all; from a point on the circle there is exactly one, of length 0.
Used in
circle theorems, belts round pulleys, constructions
The value a function gets closer and closer to as x gets closer and closer to some number, even if the function is not defined exactly there.
Precisely: The limit of f(x) as x → a is L if f(x) can be made as close to L as we like by taking x close enough to a (but not equal to it); it is written lim x→a f(x) = L.
How it works
Try values of x nearer and nearer to a, from both sides, and watch f(x). For (x² − 1) ÷ (x − 1), x = 1 gives 0 ÷ 0, but x = 0.9, 0.99, 1.01, 1.1 give 1.9, 1.99, 2.01, 2.1: the values close in on 2, so the limit is 2.
Examples
The limit of (x² − 1) ÷ (x − 1) as x → 1 is 2.
The limit of sin x ÷ x as x → 0 is 1, with x in radians.
The limit of 1 ÷ x as x grows larger and larger is 0.
The derivative is a limit: the slope of a chord as its two ends come together.
Do not confuse: The limit at a is about values near a, not the value at a; the function need not even be defined at a.
Used in
derivatives, the sum of an infinite GP, continuity in Class 12
The straight line from your eye to the thing you are looking at.
Precisely: The line drawn from the eye of an observer to the point of the object being viewed; its angle with the horizontal is the angle of elevation or depression.
How it works
Draw the horizontal line at eye level and the line of sight to the object. Together with a vertical line they make a right triangle, which trigonometry can solve.
Examples
Looking at the top of a minaret, your line of sight slants upwards from your eye to its top.
From a bridge, the line of sight to each river bank slants down below the horizontal.
An observer 1.5 m tall looks up at a chimney: the line of sight starts at her eyes, 1.5 m above the ground, not at her feet.
Do not confuse: The line of sight starts at the eye, so an observer's height must be added to what the triangle gives.
A quantity that goes down by the same amount every step, like water in a tank falling 0.5 m every month.
Precisely: A decrease described by y = ax + b with a negative slope a: equal steps in x give equal falls in y.
How it works
Find the fixed fall for each step; that is the slope, and it is negative. Its graph is a straight line going down from left to right.
Examples
Water in a tank that drops 0.5 m every month shows linear decay.
A ₹100 note spent at ₹5 a day, leaving 100 − 5n, is linear decay.
Club members leaving at 9 an hour, 120 − 9n, is another linear decay.
Do not confuse: Linear decay falls by the same amount each step; exponential decay falls by the same fraction each step (like a bouncing ball, 3/4 of its height each time).
Used in
linear relationships, graphs with negative slope, simple forecasting
OpenStax: College Algebra 2e, 4.1 Key Terms (decreasing linear function)
linear equation in two variables
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An equation like 3x + 4y = 20 with two unknowns, each only to the power 1; its graph is a straight line.
Precisely: An equation that can be written as ax + by + c = 0, where a, b, c are real numbers and a and b are not both 0.
How it works
On its own it has infinitely many solutions, one for every point on its line: choose x, then work out y. A second equation is needed to pin down one pair.
Examples
If each ride costs ₹3 and each game of Hoopla ₹4, spending ₹20 gives the linear equation in two variables 3x + 4y = 20.
x − 2y = 0 is a linear equation in two variables: (2, 1), (4, 2) and (6, 3) all satisfy it.
The linear equation in two variables y = 2x − 2 is the line through (1, 0) and (2, 2).
Do not confuse: x² + y = 4 is not linear (x is squared). And a linear equation in one variable, like 2x + 3 = 11, has just one solution.
Used in
word problems with two unknowns, straight-line graphs, pairs of equations
The set of all the points that obey one rule, such as "5 cm from A" or "equally far from A and B".
Precisely: The set of all points, and only those points, that satisfy a given condition.
How it works
Name the rule, find every point that fits it, and check that no point outside your answer fits. The locus of points at a fixed distance from one point is a circle; the locus of points equally far from two points is the perpendicular bisector of the segment joining them.
Examples
The locus of points 4 cm from a point O is the circle with centre O and radius 4 cm.
The locus of points equidistant from A and B is the perpendicular bisector of AB.
The locus of the centres of all circles through A and B is the same perpendicular bisector of AB.
The locus of the midpoints of all chords of one fixed length in a circle is a smaller circle with the same centre.
Do not confuse: A locus must contain every point that fits and no other point: finding a few points that fit is not enough.
Used in
defining a circle, constructions, conic sections (Class 11)