Grade 11 Analytical Geometry

Key concepts

• Equation of a line: y = mx + c or y − y₁ = m(x − x₁)
• Angle of inclination: tan θ = m (θ measured from the positive x-axis)
• Parallel lines: m₁ = m₂; perpendicular lines: m₁ × m₂ = −1
• Medians, altitudes and perpendicular bisectors of triangles
• Angle between two lines from their inclinations
• Distance, midpoint and gradient to prove properties of shapes

Worked examples

Find the angle of inclination of a line with gradient 1.

tan θ = 1, so θ = 45°.

Find the equation of the line through A(2; 3) parallel to y = 4x − 1.

Parallel lines have equal gradients, so m = 4. y − 3 = 4(x − 2), so y = 4x − 5.

Find the equation of the line through the origin perpendicular to y = 2x + 3.

Perpendicular gradients multiply to −1, so m = −½. The line passes through the origin, so c = 0: y = −½x.

A line has gradient −1. Find its angle of inclination.

tan θ = −1. The reference angle is 45°, and a negative gradient means an obtuse inclination: θ = 180° − 45° = 135°.

Common mistakes

• Giving a negative angle of inclination instead of adding 180°.
• Using the negative instead of the negative reciprocal for a perpendicular gradient.
• Substituting x and y the wrong way round in y − y₁ = m(x − x₁).
• Measuring the inclination from the y-axis.

Exam tips

• Sketch the points before you start; it shows you whether a gradient should be positive or negative.
• For an altitude, use the perpendicular gradient; for a median, use the midpoint.
• If tan θ is negative, the angle of inclination is 180° minus the reference angle.

Frequently asked questions

What is the angle of inclination?

The angle a line makes with the positive x-axis, measured anticlockwise. It is between 0° and 180°, and tan θ equals the gradient.

How do I find the angle between two lines?

Find each line's angle of inclination, then subtract the smaller from the larger (using the exterior angle of the triangle the lines form with the x-axis).

Is the equation of a circle in Grade 11?

No. Circles and tangents in analytical geometry are Grade 12 work.