In this topic we will learn how to create an equation of a circle with different given. Being familiar of the method will boost our speed and accuracy. Most of the problems we encounter at school are just recycled.
General equation of a circle:
where C, D and E are real numbers
Sample Problem 1: (4th Year 2013 MMC Elimination)
Find the center and the radius of the circle
To solve this problem we need to convert this standard form to center radius form by completing the square of x and y.
From the equation above we can see that h=2, k=-7 and radius of √6
The desired answer is (2,-7) and units
Sample Problem 2: (4th Year 2005 MMC elimination)
Write the equation of the circle tangent to y=-3 and with the center at (-2,2) in center radius form.
We write the equation of the circle:
, we are given already with . We just need the radius.
Solve for radius:
The radius of the circle is equal to the distance between the point (-2,2) and y=-3. Since the circle passes line y=-3 it intersects the point the point (-2,-3). It is now obvious that the radius is 5 units.
Sample Problem 3: (4th year 2005 MMC elimination)
Find the equation of the circle in standard form concentric with x2+y2+4x-8y-12=0 and passing through (2,-1)
Concentric circle are circles with the same center. We can get the center of the circle from x2+y2+4x-8y-12=0. By completing the square we get h=-2, k=4. To solve for the radius of the desired circle we solve for the distance between the center (-2,4) and a point on the circumference of the circle (2,-1) using the distance formula.
Solving for the equation of the circle: