Find the standard form of the equation of the circle having the following properties:
Center at the origin
Containing the point (-4,1)
To find the standard form of the equation of a circle, we need to know two things: the center of the circle and its radius. In this case, we know that the center is at the origin (0, 0) and the circle contains the point (-4, 1).
The general equation for a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
In our case, since the center is at the origin (0, 0), the equation of the circle becomes:
(x - 0)^2 + (y - 0)^2 = r^2
Simplifying, we have:
x^2 + y^2 = r^2
Now, to find the value of r, we can substitute the coordinates of the given point (-4, 1) into the equation.
Substituting x = -4 and y = 1 into the equation x^2 + y^2 = r^2:
(-4)^2 + (1)^2 = r^2
16 + 1 = r^2
17 = r^2
So, the equation of the circle with a center at the origin and containing the point (-4, 1) in standard form is:
x^2 + y^2 = 17
To find the standard form of the equation of a circle, we need to know the center and the radius of the circle. In this case, we are given that the center is at the origin, which is (0,0), and the circle contains the point (-4,1).
Step 1: Find the radius of the circle.
The radius can be found by using the distance formula between the center and the given point:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
d = √[(-4 - 0)^2 + (1 - 0)^2]
d = √[(-4)^2 + 1^2]
d = √[16 + 1]
d = √17
Step 2: Write the equation of the circle.
The standard form of the equation of a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
In this case, the center is at the origin (0,0), and the radius is √17. Substituting these values into the equation, we get:
(x - 0)^2 + (y - 0)^2 = (√17)^2
x^2 + y^2 = 17
So, the standard form of the equation of the circle is x^2 + y^2 = 17.
The radius of the circle must be
sqrt (4^2 + 1^2) = sqrt 17
The equation is therefore
x^2 + y^2 = 17
or, in a more standard form,
x^2/17 + y^2/17 = 1