Circles Flashcards
What are the equations for circles?
(x-a)^2 + (y-b)^2 = r^2
x^2 + y^2 + 2gx + 2fy + c = 0
How would you extract the centre and radius of a circle with the equation (x-a)^2 + (y-b)^2 = r^2 ?
Centre = (a,b) radius = square root of r^2
How would you extract the centre and radius from a circle with the equation x^2 + y^2 + 2gx + 2fy + c = 0 ?
Centre (-g,-f)
radius = square root of g^2 + f^2 - c
How would you prove that an equation does not belong to a circle?
Test the radius
If g^2 + f^2 - c < 0 then it is not a circle as negatives cannot be square rooted
How would you decide whether a point lied within, on the circumference of, or outside a circle?
Extract Centre and radius
Use the distance formula to work out the distance between the point and the centre.
If point > radius, point lies out with the circle
If point = radius, point lies on circumference
If point < radius, point lies within the circle
How would you find the equation of a circle when given two co ordinates?
Find Midpoint between the two points = Centre
Find distance between the Centre and a point = radius
Write out general formula
Sub values in
What would you do if asked which values of ‘k’ would represent a circle?
Write out general formula for a circle (x^2 + y^2 + 2gx + 2fy + c = 0)
Find what g and f are
For an equation to be a circle, g^2 + f^2 - c > 0
Sub values in
Find value for ‘k’
When do tangents appear to circles?
When they intersect once
How do you find out where a line intersects a circle?
Let y=y by substituting line equation where the y values in the circle equation are
Factorise
Solve for x
Sub x into an OG equation to find y-co ordinates
‘line meets circle at (,) (and (,))
there should always be a common factor when simplifying, if not, try again
How do you find out if a line is a tangent to a circle?
Let y=y by substituting line equation where the y values in the circle equation are
Factorise
Solve for x
Find y partner by subbing x into OG equation
‘since line meets at only one point (x, y), the line is a tangent to the circle’
How do you find out if a line is a tangent to a circle if there is an unknown coefficient?
Let y=y by substituting line equation where the y values in the circle equation are
Sub into discriminant
If discriminant = 0 then roots are real and equal, there is only one solution, line is a tangent to the circle
Else, line is not a tangent to the circle
How do you find the equation of tangents to circles?
(If asked to show that the point lies on the circle, sub co ordinate into circle equation and make sure it comes out as 0)
Find centre of the circle
Find the gradient of the radius (between Centre and given point)
Use m x m = -1 to find the gradient of the tangent
Sub tangent point and gradient into y-b=m(x-a)
What are the ways circles can intersect?
Hint : there’s five
Don’t touch (outside and far away) - d >r1 + r2
Touch externally (circumferences touch) - d = r1 + r2
Meet at two distinct points (overlap) - r1 - r2 < d < r1 + r2
Touch internally (inside one another, touching at circumference) - d = r1 - r2
Don’t touch (inside one another) - d < r1 - r2
How would you work out what type of intersection circles have?
Write the equations of both circles Extract the Centre and radius from both Calculate the distance between the two centres Add the two radii together distance <=> r1 + r2 therefore circles \_\_\_\_\_\_\_
If you need to find a point and you have the centre of another circle and an x coordinate, how would you find P?
Draw diagram
Draw triangle based on info given
Calculate length of desired side through pythagorus
Could be positive or negative
