How do you find the equation given the centre and a point on the circle?

Use r^2= (distance from centre to the point)^2 in (x-a)^2+(y-b)^2=r^2.

If AB is a diameter, where is the centre?

At the midpoint of AB; the radius is half of AB.

How do you find the centre and radius from x^2+y^2+2fx+2gy+c=0?

Complete the square: centre (-f,-g), r^2=f^2+g^2-c.

How do you find where a line meets a circle?

Substitute the line into the circle equation and solve the quadratic.

What does the discriminant tell you about a line and a circle?

0: two points; =0: one point (tangent); <0: no intersection.

When does x^2+y^2+2fx+2gy+c=0 not represent a circle?

When f^2+g^2-c\le0, because r^2 would not be positive.

Equation of a circleAQA A-Level Maths: Flashcards

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Equation of a circle, centre $(a,b)$, radius $r$?

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Equation of a circle, centre $(a,b)$ , radius $r$ ?: $(x-a)^2+(y-b)^2=r^2$
Centre and radius of $(x+4)^2+(y-1)^2=9$ ?: Centre $(-4,1)$ , radius $3$ .
Equation of a circle with centre at the origin?: $x^2+y^2=r^2$
How do you find the equation given the centre and a point on the circle?: Use $r^2=$ (distance from centre to the point) $^2$ in $(x-a)^2+(y-b)^2=r^2$ .
If $AB$ is a diameter, where is the centre?: At the midpoint of $AB$ ; the radius is half of $AB$ .
How do you find the centre and radius from $x^2+y^2+2fx+2gy+c=0$ ?: Complete the square: centre $(-f,-g)$ , $r^2=f^2+g^2-c$ .
Complete the square: $x^2-6x$ .: $(x-3)^2-9$
How do you test whether $(p,q)$ is inside the circle?: $(p-a)^2+(q-b)^2<r^2$
How do you find where a line meets a circle?: Substitute the line into the circle equation and solve the quadratic.
What does the discriminant tell you about a line and a circle?: $>0$ : two points; $=0$ : one point (tangent); $<0$ : no intersection.
How do you find where a circle meets the $y$ -axis?: Put $x=0$ and solve for $y$ .
When does $x^2+y^2+2fx+2gy+c=0$ not represent a circle?: When $f^2+g^2-c\le0$ , because $r^2$ would not be positive.

Exam questions on Equation of a circle

A circle has equation $(x-3)^2+(y+2)^2=25$ .
The line $x=3$ meets the circle at two points. Find their coordinates.2 marks
A circle has equation $x^2+y^2-6x+10y+18=0$ .
Show that the circle does not meet the $x$ -axis.2 marks
A circle has centre $C(2,-1)$ and passes through the point $P(5,3)$ .
Find the equation of the circle.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).