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18: Circles, Tangents and Chords © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

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Circle Problems Module C1 AQA Edexcel OCR MEI/OCR Module C2 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

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Circle Problems x radius Some properties of circles may be needed in solving problems. This is the 1 st one The tangent to a circle is perpendicular to the radius at its point of contact tangent Tangents to Circles A line which is perpendicular to a tangent to any curve is called a normal. For a circle, the radius is a normal.

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Circle Problems x Diagrams are very useful when solving problems involving circles e.g.1 Find the equation of the tangent at the point (5, 7) on a circle with centre (2, 3) (2, 3) (5, 7) x tangent The tangent to a circle is perpendicular to the radius at its point of contact Method: The equation of any straight line is. gradient Find m using Substitute for x, y, and m in to find c. Find using We need m, the gradient of the tangent. Tangents to Circles

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Circle Problems e.g.1 Find the equation of the tangent at the point (5, 7) on a circle with centre (2, 3) Solution: Substitute the point that is on the tangent, (5, 7): x (2, 3) (5, 7) x tangent gradient or

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Circle Problems e.g.2 The centre of a circle is at the point C (-1, 2). The radius is 3. Find the length of the tangents from the point P ( 3, 0). x C (-1, 2) Solution: P (3,0) x 3 Method: Sketch! Find CP and use Pythagoras’ theorem for triangle CPA A tangent Use 1 tangent and join the radius. The required length is AP.

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Circle Problems Exercises 1. Find the equation of the tangent at the point A(3, -2 ) on the circle 2. Find the equation of the tangent at the point A(7, 6) on the circle Ans: Solutions are on the next 2 slides Ans:

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Circle Problems 1.Find the equation of the tangent at the point A (3, -2) on the circle Solution: Centre is (0, 0). Sketch! Equation of tangent isor Gradient of radius, Gradient of tangent, gradient x x (0, 0) (3, -2) gradient m

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Circle Problems 2. Find the equation of the tangent at the point A (7, 6) on the circle Solution: Centre is (4, 2). or Gradient of tangent, Gradient of radius, gradient (4, 2) (7, 6) x tangent gradient x

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Circle Problems x Chords of Circles The perpendicular from the centre to a chord bisects the chord chord Another useful property of circle is the following:

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Circle Problems x chord The point M (4, 3) is the mid-point of a chord. Find the equation of this chord. e.g. A circle has equation Find the gradient of the radius Method: We need m and c in Complete the square to find the centre Find the gradient of the chord Substitute the coordinates of M into to find c.

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Circle Problems x chord Centre C is C Solution: The point M (4, 3) is the mid-point of a chord. Find the equation of this chord. e.g. A circle has equation Tip to save time: Could you have got the centre without completing the square?

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Circle Problems (b) x chord Exercise 1.A circle has equation (a) Find the coordinates of the centre, C. (b) Find the equation of the chord with mid- point (2, 6). Solution: (a) Centre is ( 1, 5 ) C Equation of chord is on the chord Equation of chord is

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Circle Problems x Semicircles The angle in a semicircle is a right angle diameter P Q A B The 3 rd property of circles that is useful is:

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Circle Problems x e.g. A circle has diameter AB where A is ( -1, 1) and B is (3, 3). Show that the point P (0, 0) lies on the circle. diameter A(-1, 1) B(3, 3) Method: If P lies on the circle the lines AP and BP will be perpendicular. Solution: P(0, 0) Hence and P is on the circle. Gradient of AP : Gradient of BP : So,

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Circle Problems B(-2, 4) diameter C(1, 2) A(3, 5) x Exercise 1.A, B and C are the points (3, 5), ( -2, 4) and (1, 2) respectively. Show that C lies on the circle with diameter AB. Solution: Gradient of BC Since AC and BC are perpendicular, C lies on the circle diameter AB. Gradient of AC

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Circle Problems

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The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

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Circle Problems The tangent to a circle is perpendicular to the radius at its point of contact The perpendicular from the centre to a chord bisects the chord The angle in a semicircle is a right angle Properties of Circles Diagrams are nearly always needed when solving problems involving circles. A line perpendicular to a tangent to any curve is called a normal. The radius of a circle is therefore a normal.

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Circle Problems e.g.1 Find the equation of the tangent at the point (5, 7) on a circle with centre (2, 3) Solution: Substitute the point that is on the tangent, (5, 7): x (2, 3) (5, 7) x tangent gradient or

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Circle Problems x chord Centre C is C Solution: The point M (4, 3) is the mid-point of a chord. Find the equation of this chord. e.g. A circle has equation

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Circle Problems x e.g. A circle has diameter AB where A is ( -1, 1) and B is (3, 3). Show that the point P (0, 0) lies on the circle. diameter A(-1, 1) B(3, 3) Method: If P lies on the circle the lines AP and BP will be perpendicular. Solution: P(0, 0) Hence and P is on the circle. Gradient of AP : Gradient of BP : So,.

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