NSC / Matric Analytical geometry — practice questions

1. A circle is described by a single distance condition. 2. Every point on the circumference sits the same distance from one fixed point. 3. That fixed point is the centre and the fixed distance is the radius. 4. Writing that condition with the distance formula gives the equation of the circle. _Source: Siyavula Mathematics Grade 12 (Everything Maths, CC BY 4.0), Chapter 7: Analytical geometry_

1. The angle of inclination is measured from the positive direction of the horizontal axis up to the line. 2. Drop a vertical from any point on the line to make a right angled triangle. 3. The rise is the side opposite the angle and the run is the side next to it. 4. Opposite divided by adjacent is the tangent, and rise divided by run is the gradient. 5. A negative gradient gives an obtuse angle, found by adding a straight angle to the negative result. _Source: Siyavula Mathematics Grade 12 (Everything Maths, CC BY 4.0), Chapter 7: Analytical geometry_

1. A point lies on a circle when its distance from the centre equals the radius. 2. It is easier to compare squared distances, so test whether the squared distance equals 25. 3. For the point 6 comma 0 the horizontal gap is 6 minus 2, which is 4, and 4 squared is 16. 4. The vertical gap is 0 minus minus 3, which is 3, and 3 squared is 9. 5. Sixteen plus 9 is 25, which matches the radius squared, so this point lies on the circle. 6. For 2 comma 3 the gaps are 0 and 6, giving 0 plus 36, which is 36 and too large. 7. For 0 comma 0 the gaps are 2 and 3, giving 4 plus 9, which is 13 and too small, and for 4 comma 2 the gaps are 2 and 5, giving 4 plus 25, which is 29. 8. The point 2 comma 3 is the trap for changing the sign of the y coordinate of the centre instead of subtracting it. _Source: Siyavula Mathematics Grade 12 (Everything Maths, CC BY 4.0), Chapter 7: Analytical geometry_