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Coordinate Geometry – The Circle This week the focus is on solving problems which involve circles, lines meeting circles and lines and circles intersecting.

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Coordinate Geometry – The Circle CONTENTS: Tangent to a Circle Revision Notes Example 1 Example 2 Example 3 Example 4 Assignment

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Tangent to a Circle The tangent to a circle is a straight line which meets the circle at one point only. The angle between the tangent and the radius of the circle at that point is always 90°, i.e. they are perpendicular. Coordinate Geometry – The Circle

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Revision Notes The distance from the centre of a circle to any point on the circle is the radius. If we know the coordinates of the centre of a circle and a point on the circle we can find the radius using the distance between two points formula: Distance 2 = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 Coordinate Geometry – The Circle

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Revision Notes The general equation of a circle is given as: (x – a) 2 + (y – b) 2 = r 2 where (a, b) is the centre of the circle Coordinate Geometry – The Circle

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Revision Notes If a line and a circle intersect we can find the point(s) of intersection by solving the equations simultaneously. With lines and circles we usually use the method of substitution to solve simultaneously. That is we substitute the equation of the line into the equation of the circle. Coordinate Geometry – The Circle

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Example 1: The line y = 2x + 13 touches the circle x 2 + ( y – 3) 2 = 20 at (-4, 5). Show that the radius at (-4, 5) is perpendicular to the line. Solution: Click here for solutionClick here for solution. Coordinate Geometry – The Circle

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Example 2: The line y = x – 2 intersects the circle (x + 1) 2 + (y + 3) 2 = 8 at the points A and B. Find the coordinates of A and B. Solution: Click here for solutionClick here for solution. Coordinate Geometry – The Circle

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Example 3: Show that the line 2y + x = 18 is a tangent to the circle (x – 2) 2 + (y – 3) 2 = 20 and determine the coordinates of the point of contact. Solution: Click here for solutionClick here for solution. Coordinate Geometry – The Circle

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Example 4: The point P(1, -2) lies on the circle centre (4, 6). a)Find the equation of the circle b)Find the equation of the tangent to the circle at P. Solution: Click here for solutionClick here for solution. Coordinate Geometry – The Circle

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Assignment This weeks assignment is the discussion forum “Assignment 4: Mixed Problems”. There are 6 questions from which you choose one. A maximum of 3 solutions are to be submitted for each question. Full instructions are in the forum. Deadline: 5:00pm on Monday 8 March 2010. Coordinate Geometry – The Circle

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COORDINATE GEOMETRY Summary. Distance between two points. In general, x1x1 x2x2 y1y1 y2y2 A(x 1,y 1 ) B(x 2,y 2 ) Length = x 2 – x 1 Length = y 2 – y.

COORDINATE GEOMETRY Summary. Distance between two points. In general, x1x1 x2x2 y1y1 y2y2 A(x 1,y 1 ) B(x 2,y 2 ) Length = x 2 – x 1 Length = y 2 – y.

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