Circles, lines and tangents Cambridge IGCSE Additional Mathematics revision
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In plain words
A line and a circle can meet twice, once or not at all. And a tangent has one property that solves nearly every tangent question: it is at right angles to the radius at the point where it touches.
5 things to know
- To find where a line meets a circle, substitute the line's equation into the circle's and solve the quadratic.
- Two different roots: the line cuts the circle at two points, and the part inside is a chord. Equal roots: the line is a tangent. No real roots: the line misses the circle.
- A tangent is perpendicular to the radius at the point of contact.
- To find the equation of the tangent at a point: find the gradient of the radius to that point, take the negative reciprocal, and use it with the point.
- The line from the centre perpendicular to a chord cuts the chord in half.
Worked example
Find the equation of the tangent to the circle (x − 1)² + (y − 2)² = 25 at the point (4, 6).
- The centre is (1, 2). Gradient of the radius to (4, 6) = (6 − 2) ÷ (4 − 1) = 4/3.
- The tangent is perpendicular to the radius, so its gradient is −3/4.
- y − 6 = −(3/4)(x − 4), which gives 3x + 4y = 36.
Worked example
Show that the line y = x + 4 is a tangent to the circle x² + y² = 8.
- Substitute: x² + (x + 4)² = 8, so 2x² + 8x + 8 = 0, which is x² + 4x + 4 = 0.
- The discriminant is 16 − 16 = 0, so the roots are equal.
- The line meets the circle at one point only, so it is a tangent.
Tips and tricks
- No calculus is needed, or expected, for tangents to circles. Use the radius.
- Before finding a tangent, check that the point really is on the circle by putting it into the equation.
It lands in your notebook with its questions as flashcards.
Circles, lines and tangents: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
What is the angle between a tangent and the radius at the point of contact?
A tangent is perpendicular to the radius.
A line is substituted into a circle and the quadratic has two different real roots. What is the line?
The part inside the circle is a chord.
The radius to a point on a circle has gradient 2. What is the gradient of the tangent there?
The negative reciprocal.
A circle has centre (0, 0). What is the gradient of the tangent at (0, 5)?
The radius is vertical, so the tangent is horizontal.
Where does the line y = x + 1 meet the circle x² + y² = 25?
x² + x − 12 = 0 gives x = 3 or −4.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
2 questions, 7 marks. Write your answers on paper, then check them.
Circles, lines and tangents
Cambridge IGCSE Additional Mathematics 0606 · 7 marks · papermunch.org
Name ______________________________ Date ______________
Find the equation of the tangent to the circle x² + y² = 25 at the point (3, −4).[4]
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3x − 4y = 25. The radius has gradient −4/3, so the tangent has gradient 3/4, and y + 4 = (3/4)(x − 3).
Determine whether the line y = x + 6 meets the circle x² + y² = 9.[3]
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It does not. Substituting gives 2x² + 12x + 27 = 0, whose discriminant is 144 − 216 = −72, which is negative.
Answers: Circles, lines and tangents
- 1. 3x − 4y = 25. The radius has gradient −4/3, so the tangent has gradient 3/4, and y + 4 = (3/4)(x − 3).
- 2. It does not. Substituting gives 2x² + 12x + 27 = 0, whose discriminant is 144 − 216 = −72, which is negative.