Problem Solving

The center of the circle is at point (0,6). If the distance between the two points
where the circle intersects the x-axis is 16, what is the area of the circle?

Is it 36pie? I'm thinking that the radius must be 6 since the center is six points up the y-axis, but that sounds too simple.

I don't think the answer is 36pi it is only the centre. To find the area , we have to determine the radius, which according to the Q stem can be derived from the two points at which the circle intersects at x axis, say (x,0) and (-x,0). The distance given between the two points is 16,
sqrt((2x)^2)=16 ===>x=2 or -2
Therefore, the two points are (2,0) and (-2,0). The radius will be then distance between (0,6)and (2,0)
or (0,6) and (-2,0) which gives sqrt(40). The area will then be 40pi.

But I am not sure whether this is the correct answer or the way to derive it. Need confirmation. Thanks.

Could someone please help confirm the answer. Thanks.

The center is at (0,6). The circle formed around this center has to be symmetrical against the Y-axis.

The distance between the two points intersecting X-axis is 16. So, we can say the length of the chord is 16.

The distance of any of the points from origin (0,0) is 8 as the circle is symmetric on Y-axis. (i.e the two intersecting points are (8,0) and (-8,0))

Now the radius = sqrt(6^2 + 8^2) = 10

Thus, the area = 100 * pie

if you draw a triangle using center O and two points of intersection call it A and B.
then you get a isoceles triangle since both OA and OB are radius so OA=OB.
And since center is 0,6 which means - Y axis divides AB symterically at 90 degress using pythagorous theorum (hyp) OA=sqrt(6^2 + 8^2) = 10

Area= Pie *10^2=100pie

@ sampath, why is the answer not 16 pie?
whats the OA?

@wtp: A and B are two points where circle intersects with x axis and since they are at the circumfrence of the circle and O is the center then OA = OB = radius of the circle.

doing a bit of construction we have triangles with 6,8 and r as sides.

thus r=10 hence A = pi* 100.

SO answer comes to 292pi.

Can somebody verify this Please?

worldpeace93 wrote:

SO answer comes to 292pi.

Can somebody verify this Please?

Hi,
You are calculating r^2 = 6^2 + 16^2 instead of r^2 = 6^2 + 8^2.

worldpeace93 wrote:

SO answer comes to 292pi.

Can somebody verify this Please?