Analytic geometry: Class 12 Mathematics

Key points

Analytic (coordinate) geometry studies geometric figures using coordinates and equations. A point in the plane is written (x, y), where x is the distance along the x-axis and y along the y-axis from the origin.

Distance formula: the distance between P(x₁, y₁) and Q(x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²]. It follows from the theorem of Pythagoras. The distance is never negative.

Midpoint formula: the midpoint of PQ is ((x₁ + x₂)/2, (y₁ + y₂)/2). Section formula: the point R dividing PQ internally in the ratio m : n is ((m x₂ + n x₁)/(m + n), (m y₂ + n y₁)/(m + n)).

Slope (gradient) of a line through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁), if x₂ ≠ x₁. Also m = tan θ, where θ is the angle the line makes with the positive x-axis. A horizontal line has slope 0. A vertical line has no slope (undefined).

Conditions for two lines with slopes m₁ and m₂:

• Parallel lines: m₁ = m₂.
• Perpendicular lines: m₁ m₂ = −1.
• Three points are collinear if the slopes of two pairs of them are equal.

Forms of the equation of a straight line:

• Slope-intercept form: y = mx + c, where c is the y-intercept.
• Point-slope form: y − y₁ = m(x − x₁).
• Two-point form: (y − y₁)/(y₂ − y₁) = (x − x₁)/(x₂ − x₁).
• Intercept form: x/a + y/b = 1, where a and b are the x-intercept and y-intercept.
• General form: ax + by + c = 0, with slope −a/b if b ≠ 0.
• Horizontal line: y = k. Vertical line: x = h.

To draw a line, find two points. The simplest are the intercepts: put y = 0 to find the x-intercept, and x = 0 to find the y-intercept. To find where two lines meet, solve their equations together as simultaneous equations.

Angle between two lines: if θ is the angle between lines of slopes m₁ and m₂, then tan θ = |(m₂ − m₁)/(1 + m₁ m₂)|. If 1 + m₁ m₂ = 0 the lines are perpendicular and θ = 90°.

Distance from a point P(x₁, y₁) to the line ax + by + c = 0 is d = |a x₁ + b y₁ + c|/√(a² + b²). The distance between two parallel lines ax + by + c₁ = 0 and ax + by + c₂ = 0 is |c₁ − c₂|/√(a² + b²).

Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. If this is zero the three points lie on one line.

Concurrent lines: three lines are concurrent if they pass through one point. Find the meeting point of two lines and check whether it satisfies the third. A family of lines through the meeting point of L₁ = 0 and L₂ = 0 can be written L₁ + k L₂ = 0.

Equations of the medians, altitudes and perpendicular bisectors of a triangle are found from these tools. The perpendicular bisector of PQ passes through the midpoint of PQ and has slope −1/m, where m is the slope of PQ. An altitude from a vertex has slope −1/m, where m is the slope of the opposite side.

Study tip: Draw a quick sketch of every problem and plot the points, because many errors show up at once. Learn the slope conditions and the point-to-line formula exactly. Always check a found equation by putting a given point into it. Show each substitution step in numerical questions.

Definitions and flashcards

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Short questions with model answers

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Common mistakes

• Writing the slope as (x₂ − x₁)/(y₂ − y₁), upside down, or taking the differences in different orders for top and bottom.
• Using m₁ = m₂ for perpendicular lines, or forgetting the negative sign in m₁ m₂ = −1.
• Forgetting the modulus signs in the distance-to-a-line and angle formulas, which gives a negative distance.
• Not writing the line in the form ax + by + c = 0 before using the point-to-line distance formula.
• Reading the slope of ax + by + c = 0 as a/b instead of −a/b.

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