WEBVTT 00:00:00.500 --> 00:00:03.868 Welcome to GCSE Edexcel Maths revision. 00:00:04.018 --> 00:00:07.689 Topic M 21: Circle equations and tangents. 00:00:07.839 --> 00:00:10.015 This video covers Higher tier. 00:00:12.667 --> 00:00:19.429 A circle with centre at the origin and radius r has equation x squared plus y squared equals r squared . 00:00:19.579 --> 00:00:24.203 It follows from Pythagoras applied to the horizontal and vertical coordinates. 00:00:24.353 --> 00:00:27.638 Radius perpendicular to a tangent at three four 00:00:32.300 --> 00:00:41.685 Worked example: x squared plus y squared equals 25 has centre open bracket 0, 0 close bracket and radius 5, not 25. 00:00:44.367 --> 00:00:56.695 A point is on the circle when its coordinates satisfy the equation. open bracket 3, 4 close bracket is on x squared plus y squared equals 25 because 9 plus 16 equals 25. 00:00:59.367 --> 00:01:06.577 If x squared plus y squared is smaller than r squared , the point lies inside the circle; if larger, it lies outside. 00:01:09.233 --> 00:01:19.525 Given x on the circle, y equals plus or minus square root of open bracket r squared minus x squared close bracket , where given x given is less than or equal to r. 00:01:19.675 --> 00:01:22.735 Most vertical lines through the circle meet it twice. 00:01:25.400 --> 00:01:30.211 A tangent touches a circle at one point and is perpendicular to the radius there. 00:01:30.361 --> 00:01:35.193 Find the radius's gradient, then take the negative reciprocal for the tangent. 00:01:37.867 --> 00:01:45.735 Worked example: At open bracket 3, 4 close bracket on the radius-5 circle, the radius gradient is 4 over 3 . 00:01:45.885 --> 00:01:48.881 The tangent gradient is minus 3 over 4 . 00:01:51.533 --> 00:02:00.012 Using y equals mx plus c, 4 equals minus 3 over 4 multiplied by 3 plus c gives c equals 25 over 4 . 00:02:00.162 --> 00:02:07.958 The tangent is y equals minus 3 over 4 x plus 25 over 4 , or 3 x plus 4 y equals 25. 00:02:10.633 --> 00:02:17.950 At open bracket r, 0 close bracket , the radius is horizontal and the tangent is vertical: x equals r. 00:02:18.100 --> 00:02:23.651 At open bracket 0, r close bracket , the tangent is horizontal: y equals r. 00:02:23.801 --> 00:02:26.486 Avoid attempting a reciprocal of zero. 00:02:29.167 --> 00:02:36.195 To check a tangent equation, confirm it passes through the stated point and its gradient is perpendicular to the radius. 00:02:38.867 --> 00:02:43.495 Substitute a line equation into the circle equation to find intersections. 00:02:43.645 --> 00:02:48.349 A tangent produces a repeated root, representing a single contact point. 00:02:51.000 --> 00:03:01.015 Worked example: For y equals 3 on x squared plus y squared equals 25, x squared plus 9 equals 25 gives x equals plus or minus 4. 00:03:01.165 --> 00:03:07.554 The two points are open bracket minus 4, 3 close bracket and open bracket 4, 3 close bracket . 00:03:10.233 --> 00:03:16.162 Circle equations describe a full locus, not one function y of x unless a branch is selected. 00:03:16.312 --> 00:03:19.862 Keep both branches when the question concerns the whole circle. 00:03:22.533 --> 00:03:25.243 That completes Circle equations and tangents. 00:03:25.393 --> 00:03:28.738 Revisit the notes and test yourself on the revision website.