Question

8 A circle passes through the points (0, 2) and (0,8) and its centre lies on the line
x=4.
Draw a sketch and find the equation of the circle.

Answer 1

Step-by-step explanation:

A circle that passes through point 0,2 and (0,8) and has a center that lies on x=4 is what given.

So we know is center (4,p) for some number p,

Since the points on circumference of the circle have the same x value, then we know that the y coordinate of the center must be the midpoint of the circumferences' y coordinates.

Why?

A circle is not a function since it doesn't pass the vertical line test. And the standard form of a circle is

${x}^{2} + {y}^{2} = {r}^{2}$

With center (0,0).

Consider the equation

${x}^{2} + {y}^{2} = 9$

The radius is 3 so two points with x coordinate 0 is

(0,3) and (0,-3). And the midpoint of them is (0,0) which is the center. Which is the defining trait of all circles.

Now, back on hand, our the midpoint of 2 and 8 is 5 so our center is

(4,5).

Now, we must find the radius. We use distance formula.

For (4,5) and (0,8).

$d = \sqrt{(8 - 5) {}^{2} + ({0 - 4) {}^{2} } }$

$d = \sqrt{3 {}^{2} + {4}^{2} }$

$d = \sqrt{25}$

$d = 5$

So our radius is 5.

So now we use the equation of center h,k.

$(x - h) {}^{2} + (y - k) {}^{2} = {r}^{2}$

H is 4

K is 5

$(x - 4) {}^{2} + (y - 5) {}^{2} = 25$

Above is sketch of graph.