Chapter 6 Circles - Year 1 Flashcards
What is important about perpendicular bisector s of a circle?
It passes through the middle of the circle therefore the midpoint of a perpendicular bisector will be the centre of a circle
What is the equation of a circle?
If you know the center of a circle and 1 point on the edge how do you calculate the radius?
You can use pythagorus as you can work out the lateral and vertical values
What is a common mistake made when working out the radius of a circle?
If you have used 2 points on the edge of a circle you need to divide your answer by 2 to get half which is the radius
What changes with the values of the center of the circle and the equation?
The sign flips (as the equation is x-a and y-b
If you are given 2 points on the edge of a circle how do you work out the center of the circle?
Find the midpoint of those points (then reverse the signs to get the first part of the equation
Then find the difference between the 2 points to find the length of the diameter (then divide it by 2)
Then square that to get the radius squared
Then fill in the equation
How can you find the center and radius of a circle if the equation of the circle is in it’s expanded form?
You can complete the square
Then bring everything without an x and y to the right side then square root it to find the radius
The center will be from the squared brackets (opposite signs)
What do you use to show that a line and a circle never meet?
The discriminant
What is special about the tangent?
It only touches the circle at 1 point
It is perpendicular to the radius
How can we use a chord to get the center of a circle?
(As long as you already know the y co ordinate
Find the perpendicular bisector
What is special about chords in circles?
If you find the perpendicular bisector of it you can find the center of the circle
How do you find the perpendicular bisector of a chord?
You find the midpoint of it
You find the m of and then take the negative reciprocal so you now know the m of the perpendicular bisector
You can then use y-y1=m(x-x1) to find the equasion
What is the significance of knowing that the perpendicular bisector chord passes through the center?
If you have 2 chords and hence found 2 bisectors you can pin point the center of the circle
How to you verify that a point lies on a circle?
You substitute the values into the circle and see if the 2 sides say equal
If you are given a an equation of a circle and you are told the gradient of a tangent how do you work out he equations of the tangents?
State the m of the radius (negative reciprocal)
Substitute that equation into the equation of the circle
Then substitute thise answers into y-y1=m(x-x1) to get the equations of the triangles
What is meant if a triangle inscribes a circle?
What is the word in term of the circle?
All 3 points of the triangle are on the circumference of the circle
The circle circumscribes the triangle
What are the 2 methods to show that a line is a diameter?
Method 1 - Using pythagorus
You prove that the sum of the squares of 2 side lengths equal the square of the diameter length
Method 2 - You show that the other 2 lines are perpendicular
Calculate the m’s of the other 2 lines
Multiply them to get -1
Then state AC is perpendicular to BC as their gradients have a product of -1
Therefore AB is the diameter
How do you find the equation of a circle when you are given 3 points which are on it’s circumference?
Find the perpendicular bisector of 2 chords
Then put them equal to each other and solve for x and then get your y value
You now know the center of the circle (swap the signs)
Then you work out the diameter of the circle and divide your answer by 2 to get the diameter
Then square it and fill in your equation of the circle
