Higher Maths 2 4 Circles1. Distance Between Two Points 2Higher Maths 2 4 Circles The Distance Formula d =d = ( y2 – y1)²( y2 – y1)² + ( x2 – x1)²( x2.

Slides:



Advertisements
Similar presentations
Coordinate Geometry Locus II
Advertisements

Coordinate Geometry Locus I
Problem Set 2, Problem # 2 Ellen Dickerson. Problem Set 2, Problem #2 Find the equations of the lines that pass through the point (1,3) and are tangent.
Digital Lesson on Graphs of Equations. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graph of an equation in two variables.
Notes Over 10.3 r is the radius radius is 4 units
Coordinate Geometry – The Circle This week the focus is on solving problems which involve circles, lines meeting circles and lines and circles intersecting.
The Circle (x 1, y 1 ) (x 2, y 2 ) If we rotate this line we will get a circle whose radius is the length of the line.
1 OBJECTIVES : 4.1 CIRCLES (a) Determine the equation of a circle. (b) Determine the centre and radius of a circle by completing the square (c) Find the.
Circles Write an equation given points
Scholar Higher Mathematics Homework Session
Revision - Quadratic Equations Solving Quadratic equations by the Quadratic Formula. By I Porter.
Higher Maths Question Types. Functions & Graphs TYPE questions (Trig, Quadratics) Sketching Graphs Composite Functions Steps : 1.Outside function stays.
Circles, Line intersections and Tangents © Christine Crisp Objective : To find intersection points with straight lines To know if a line crosses a circle.
EXAMPLE 1 Graph an equation of a circle
EXAMPLE 1 Graph an equation of a circle Graph y 2 = – x Identify the radius of the circle. SOLUTION STEP 1 Rewrite the equation y 2 = – x
Definitions  Circle: The set of all points that are the same distance from the center  Radius: a segment whose endpoints are the center and a point.
Formulas Things you should know at this point. Measure of an Inscribed Angle.
10.6 Equations of a Circle Standard Equation of a Circle Definition of a Circle.
EXAMPLE 1 Write an equation of a circle Write the equation of the circle shown. The radius is 3 and the center is at the origin. x 2 + y 2 = r 2 x 2 +
Pupils notes for Circle Lessons.  The equation of a circle with centre ( a, b ) and radius r is We usually leave the equation in this form without multiplying.
Circles Revision Transformations Transformations Intercepts Intercepts Using the discriminant Using the discriminant Chords Chords.
1Higher Maths Quadratic Functions. Any function containing an term is called a Quadratic Function. The Graph of a Quadratic Function 2Higher Maths.
Higher Maths Strategies Click to start The Circle.
EXAMPLE 1 Graph an equation of a circle
CIRCLE EQUATIONS The Graphical Form of the Circle Equation Inside, Outside or On the Circle Intersection Form of the Circle Equation Find intersection.
Quadratic Functions 2.
Module :MA0001NP Foundation Mathematics Lecture Week 9
EXAMPLE 1 Graph the equation of a translated circle
Higher Outcome 4 Higher Unit 2 The Graphical Form of the Circle Equation Inside, Outside or On the Circle.
Do Now : Evaluate when x = 6, y = -2 and z = The Quadratic Formula and the Discriminant Objectives Students will be able to: 1)Solve quadratic.
EXAMPLE 3 Write the standard equation of a circle The point (–5, 6) is on a circle with center (–1, 3). Write the standard equation of the circle. SOLUTION.
Unit 1 – Conic Sections Section 1.2 – The Circle Calculator Required.
5.2 Graph and Write Equations of Circles Pg180.  A circle is an infinite set of points in a plane that are equal distance away from a given fixed point.
Circles in the Coordinate Plane I can identify and understand equations for circles.
Circles Students will be able to transform an equation of a circle in standard form to center, radius form by using the complete the square method.
Circles 5.3 (M3). EXAMPLE 1 Graph an equation of a circle Graph y 2 = – x Identify the radius of the circle. SOLUTION STEP 1 Rewrite the equation.
Coordinate Geometry Locus III - Problems By Mr Porter.
TANGENCY Example 17Prove that the line 2x + y = 19 is a tangent to the circle x 2 + y 2 - 6x + 4y - 32 = 0, and also find the point of contact. ********
Section 6.2 – The Circle. Write the standard form of each equation. Then graph the equation. center (0, 3) and radius 2 h = 0, k = 3, r = 2.
10-3 Circles Learning Target: I can use equations of circles to model and solve problems Goal 2.09.
Distance and Midpoint Intercepts Graphing Lines Graphing Circles Random.
The Equation of a Circle. Aims To recognise the equation of a circle To find the centre and radius of a circle given its equation To use the properties.
8.1 The Rectangular Coordinate System and Circles Part 2: Circles.
EXAMPLE 3 Use the quadratic formula y = 10x 2 – 94x = 10x 2 – 94x – = 10x 2 – 94x – 300 Write function. Substitute 4200 for y. Write.
Circles equations. Find the missing value to complete the square. 6.x 2 – 2x +7. x 2 + 4x +8. x 2 – 6x + Circles – Warm Up Find the missing value to complete.
Circles Formula. x 2 + y 2 = r 2 Formula for Circle centered at the origin Center point (0,0) Radius = r.
Concept. Example 1 Write an Equation Given the Radius LANDSCAPING The plan for a park puts the center of a circular pond of radius 0.6 mile at 2.5 miles.
Solve this equation Find the value of C such that the radius is 5.
EXAMPLE 1 Write an equation of a circle Write the equation of the circle shown. SOLUTION The radius is 3 and the center is at the origin. x 2 + y 2 = r.
HIGHER MATHEMATICS Unit 2 - Outcome 4 The Circle.
Then/Now You wrote equations of lines using information about their graphs. Write the equation of a circle. Graph a circle on the coordinate plane.
Warm Up Find the slope of the line that connects each pair of points. – (5, 7) and (–1, 6) 2. (3, –4) and (–4, 3)
10-8 Equations of Circles 1.Write the equation of a circle. 2.Graph a circle on the coordinate plane.
S TARTER Can you write an equation that will produce a circle radius 6? Can you write an equation that will produce a circle radius 2? What is the centre.
SECTIONS Lecture 3 of CIRCLES CONIC.
Unit 2 revision Q 1 How do you find the root of an equation between two given values to 1 dp ? 2.1.
The exact values can be found by solving the equations simultaneously
Circle geometry: Equations / problems
Solving harder linear Simultaneous Equations
Warm-up Solve using the quadratic formula: 2x2 + x – 5 =0
Solving Quadratic Equations by the Quadratic Formula
Circle Centre (a, b) radius r
WEEK 1 HIGHER.
r > 0 Does circles touch externally or internally ? radius = (a,b)
The Circle x2+y2+2gx+2fy+c = 0 (x-a)2 + (y-b)2 = r2 x2 + y2 = r2
Solve Simultaneous Equations One Linear, one quadratic [Circle]
Objectives Write equations and graph circles in the coordinate plane.
The equation of a circle is based on the Distance Formula and the fact that all points on a circle are equidistant from the center.
Warm Up #4 1. Write 15x2 + 6x = 14x2 – 12 in standard form. ANSWER
Presentation transcript:

Higher Maths 2 4 Circles1

Distance Between Two Points 2Higher Maths 2 4 Circles The Distance Formula d =d = ( y2 – y1)²( y2 – y1)² + ( x2 – x1)²( x2 – x1)² √ B ( x 2, y 2 ) A ( x 1, y 1 ) y 2 – y 1 x 2 – x 1 Example Calculate the distance between (-2,9) and (4,-3). d =d = + 6²6² √ 12 ² = 180 √ = 5 √ 6 Where required, write answers as a surd in its simplest form.

Points on a Circle 3Higher Maths 2 4 Circles Example Plot the following points and find a rule connecting x and y. ( 5, 0 )( 4, 3 )( 3, 4 )( 0, 5 ) (-3, 4 )(-4, 3 )(-5, 0 )(-4,-3 ) (-3,-4 )( 0,-5 )( 3,-4 )( 4,-3 ) All points lie on a circle with radius 5 units and centre at the origin. x ² + y ² = 25 x ² + y ² = r ² For any point on the circle, For any radius...

The Equation of a Circle with centre at the Origin 4Higher Maths 2 4 Circles x ² + y ² = r ² For any circle with radius r and centre the origin, The ‘Origin’ is the point (0,0) origin Example Show that the point ( - 3, ) lies on the circle with equation 7 x ² + y ² = 16 x ² + y ²x ² + y ² = ( -3 ) ² + ( ) ² 7 = = 16 Substitute point into equation: The point lies on the circle.

The Equation of a Circle with centre ( a, b ) 5Higher Maths 2 4 Circles ( x – a ) ² + ( y – b ) ² = r ² For any circle with radius r and centre at the point ( a, b )... Not all circles are centered at the origin. ( a, b )( a, b ) r Example Write the equation of the circle with centre ( 3, -5 ) and radius 2 3. ( x – a ) ² + ( y – b ) ² = r ² ( x – 3 ) ² + ( y – ( -5 ) ) ² = ( ) ² 2 3 ( x – 3 ) ² + ( y + 5 ) ² = 12

6Higher Maths 2 4 Circles The General Equation of a Circle ( x + g ) 2 + ( y + f ) 2 = r 2 ( x g x + g 2 ) + ( y fy + f 2 ) = r 2 x 2 + y g x + 2 f y + g 2 + f 2 – r 2 = 0 x 2 + y g x + 2 f y + c = 0 c = g 2 + f 2 – r 2 r 2 = g 2 + f 2 – c r = g 2 + f 2 – c Try expanding the equation of a circle with centre ( - g, - f ). General Equation of a Circle with center ( - g, - f ) and radius r = g 2 + f 2 – c this is just a number...

7Higher Maths 2 4 Circles Circles and Straight Lines A line and a circle can have two, one or no points of intersection. r A line which intersects a circle at only one point is at 90° to the radius and is is called a tangent. two points of intersection one point of intersection no points of intersection

8Higher Maths 2 4 Circles Intersection of a Line and a Circle Example Find the intersection of the circle and the line 2 x – y = 0 x 2 + ( 2 x ) 2 = 45 x x 2 = 45 5 x 2 = 45 x 2 = 9 x = 3 or -3 y = 2 x x 2 + y 2 = 45 Substitute into y = 2 x : How to find the points of intersection between a line and a circle: rearrange the equation of the line into the form y = m x + c substitute y = m x + c into the equation of the circle solve the quadratic for x and substitute into m x + c to find y y = 6 or -6 Points of intersection are ( 3,6 ) and ( -3,-6 ).

9Higher Maths 2 4 Circles Intersection of a Line and a Circle (continued) Example 2 Find where the line 2 x – y + 8 = 0 intersects the circle x 2 + y x + 2 y – 20 = 0 x 2 + ( 2 x + 8 ) x + 2 ( 2 x + 8 ) – 20 = 0 x x x x + 4 x + 16 – 20 = 0 5 x x + 60 = 0 5 ( x x + 12 ) = 0 5 ( x + 2 )( x + 6 ) = 0 x = -2 or -6 Substituting into y = 2 x + 8 points of intersection as ( -2,4 ) and ( -6,-4 ). Factorise and solve

10Higher Maths 2 4 Circles The Discriminant and Tangents x =x = -b-b b 2 – ( 4 ac ) ± 2 a2 a Discriminant The discriminant can be used to show that a line is a tangent: substitute into the circle equation rearrange to form a quadratic equation evaluate the discriminant y = m x + c b 2 – ( 4 ac ) > 0 Two points of intersection b 2 – ( 4 ac ) = 0 The line is a tangent b 2 – ( 4 ac ) < 0 No points of intersection r

11Higher Maths 2 4 Circles Circles and Tangents Show that the line 3 x + y = -10 is a tangent to the circle x 2 + y 2 – 8 x + 4 y – 20 = 0 Example x 2 + (- 3 x – 10 ) 2 – 8 x + 4 (- 3 x – 10 ) – 20 = 0 x x x – 8 x – 12 x – 40 – 20 = 0 10 x x + 40 = 0 b 2 – ( 4 ac ) = 40 2 – ( 4 × 10 × 40 ) = 0 = 1600 – 1600 The line is a tangent to the circle since b 2 – ( 4 ac ) = 0

12Higher Maths 2 4 Circles Equation of Tangents To find the equation of a tangent to a circle: Find the center of the circle and the point where the tangent intersects Calculate the gradient of the radius using the gradient formula Write down the gradient of the tangent Substitute the gradient of the tangent and the point of intersection into y – b = m ( x – a ) Straight Line Equation y – b = m ( x – a ) m tangent = –1 m radius x2 – x1x2 – x1 y2 – y1y2 – y1 m radius = r