The circle: equation, line and circle, tangent
Build and read the equation of a circle, find where a line meets a circle, and work with the tangent and its length.
π Start here: read the steps of the βLessonβ in order (1, 2, 3 β¦) and answer each question. Then continue with βPracticeβ.
- UK: Year 10
- France: grade 11
- Iran: grade 12
- Afghanistan: grade 10
- Syria: grade 10
- Ukraine: grade 9
- Russia: grade 9
Guide
A circle is the set of points at a fixed distance r (the radius) from a fixed point (the centre). With centre C(h,k) the equation is (x β h)2 + (y β k)2 = r2. The signs of h and k come out the opposite way in the equation: the centre (3, β2) gives (x β 3)2 + (y + 2)2.
The general form x2 + y2 + 2gx + 2fy + c = 0 has centre (βg, β f) and radius r = β(g2 + f2 β c). To find where a line meets a circle, substitute y from the line into the circle and solve the quadratic: two solutions, two points; one solution, a tangent; none, the line misses the circle.
A tangent is perpendicular to the radius at the point of contact. The length of the tangent from an outside point P to the point of contact is β(OP2 β r2), because the triangle OPT has a right angle at T.