Grade 12 Analytical Geometry Questions and Answers

Grade 12 analytical geometry questions and answers for CAPS Paper 2: distance, midpoint, gradients, equations of circles and tangents, fully worked.

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Analytical geometry carries about 40 of the 150 marks in Paper 2. Most questions use a small set of formulas from the information sheet, so a neat sketch and careful substitution earn most of the marks.

Question 1: Distance and midpoint

For A(−2; 3) and B(4; −5), find AB and the midpoint of AB. AB = √((4 + 2)² + (−5 − 3)²) = √(36 + 64) = 10. M = ((−2 + 4)/2; (3 − 5)/2). Answer: AB = 10 units and M(1; −1).

Question 2: Perpendicular line

Find the equation of the line through (2; 5) perpendicular to y = 2x + 1. Perpendicular gradients multiply to −1, so m = −½. Then y − 5 = −½(x − 2). Answer: y = −½x + 6.

Question 3: Angle of inclination

Find the angle of inclination of y = √3x + 2. The gradient is √3 and tan θ = m, so tan θ = √3. Answer: θ = 60°.

Question 4: Equation of a circle

Find the equation of the circle with centre (3; −2) passing through (6; 2). r² = (6 − 3)² + (2 + 2)² = 9 + 16 = 25. Answer: (x − 3)² + (y + 2)² = 25.

Question 5: Centre and radius

Find the centre and radius of x² + y² − 6x + 4y − 12 = 0. Complete the square: (x − 3)² + (y + 2)² = 12 + 9 + 4 = 25. Answer: centre (3; −2) and radius 5.

Question 6: Tangent to a circle

Find the tangent to x² + y² = 25 at (3; 4). The radius has gradient 4/3, and a tangent is perpendicular to the radius, so the tangent's gradient is −¾. Then y − 4 = −¾(x − 3). Answer: y = −¾x + 25/4.

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Frequently asked questions

Are the analytical geometry formulas given in the exam?

Yes, the distance, midpoint, gradient and circle formulas are on the information sheet, but you need to know when to use each one.

Why does a tangent question need the radius?

A tangent is perpendicular to the radius at the point of contact, so the radius gives you the gradient you need.

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