geometry tough questn

[sukh]

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?

[shankar.ashwin]

Best way to solve this would be to draw the circle. You know the centre is (0,6) And the length of a chord is given as 16.

Now since the circle is symmetric about the Y-axis, half the length of the chord would be 8. Now complete the right triangle,

6,8 and 10 are pythagoras triplets. The radius would be 10 and the area 100Pi

[neelgandham]

Please refer to the attachment, which has the diagram for shankar's explanation!

My 2 cents

Let O be the centre of the circle(0,6)
let D be the origin (0,0)

Using the property 'The line segment drawn from the centre to the midpoint of the chord is perpendicular to the chord', We know that if BD=DC=8 then Angle ODC = 90 degrees.

Implies, ODC = 90 degrees, OD=6 units, DC = 8units.6,8 and 10 are pythagoras triplets. The radius would be 10 and the area 100Pi