Circles and lines in analytic geometry · Year 12
In Circles and lines in analytic geometry (Year 12) you practise 5 skills: A line and a circle: cut, touch or miss · Where a line meets a circle · The tangent to a circle · A circle touching the axes · Chords of a circle. A circle is all the points at a fixed distance from one point, its centre. By the distance formula, a point is on the circle when the squares of the differences between its coordinates and the centre’s add up to the square of the radius, and that is the circle’s equation.
A line and a circle meet at two points, at one point or not at all. Compare the distance from the centre to the line with the radius: when the distance is less than the radius the line cuts the circle, when it equals the radius the line touches the circle, and when it is more than the radius they do not meet. Another way is to put the line’s equation into the circle’s and solve: the number of solutions is the number of points where they meet. The tangent to a circle at a point is perpendicular to the radius to that point, so its gradient is the negative reciprocal of the radius’s gradient. A circle touches an axis when the distance from its centre to the axis equals the radius. The perpendicular from the centre of a circle to a chord cuts the chord in half. So half the chord, the perpendicular and the radius to an end of the chord make a triangle with a right angle, and Pythagoras’ theorem gives the length of the chord.
Tip: The tangent is perpendicular to the radius where it touches the circle.
What you learn
A line and a circle: cut, touch or miss
For example: Here are the circle (x + 3)² + (y − 4)² = 4 and the line 12x − 5y = −95 . How far is the centre of the circle from the line?
Answer and solution
The answer: 3
The centre is (−3, 4) . The left side gives −56 , and the difference from the right side is −56 − (−95) = 39 . The square root of the sum of the squared coefficients is √(144 + 25) = 13 , so the distance is 39 ÷ 13 = 3 .
Where a line meets a circle
For example:
The line y = −3 cuts the circle x² + y² = 25 . What are the values of x where they meet?Answer and solution
The answer: x = −4 or x = 4
Put it in: x² + (−3)² = 25 , so x² = 16 , that is x = 4 or x = −4 .
The tangent to a circle
For example:
The point A(4, −3) is on the circle x² + y² = 25 . What is the gradient of the tangent to the circle at A ?Answer and solution
The answer: 4/3
The radius OA has gradient −3 ÷ 4 = −3/4 , so the tangent has gradient 4/3 .
A circle touching the axes
For example: A circle with centre (−5, 9) touches the x axis. What is its radius?
Answer and solution
The answer: 9
The centre is as far from the x axis as the absolute value of its y coordinate, so the radius is 9 .
Chords of a circle
For example:
The points A(−3, 8) and B(1, 0) are on the circle (x + 3)² + (y − 3)² = 25 . Where is the midpoint of the chord AB ?Answer and solution
The answer: (−1, 4)
The x coordinate: (−3 + 1) ÷ 2 = −1 , and the y coordinate: (8 + 0) ÷ 2 = 4 , so the midpoint is (−1, 4) .
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows selected AS and A level mathematics practice in England (the GCE subject content).