If the circle in the figure above is centered at the origin of the coordinate axes, which of the following coordinates represents a point that lies on the circle?

A. (3,4)
B. (5,5)
C. (1,9)
D. (8,6)
E. (6,6)


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Distance from 0 to the the y intercept of the circle is 10. So pick the coordinates whos squares add upto a 100. Since root of 100 = distance from origin = 10.

From the given figure we have radius = 10
so the equation of the given circle is \(x^2 + y^2 = 10^2\)
only (8,6) satisfies the equation

Hence option D is correct

lostnumber , if by "arc" you mean the circumference line, no.
A point lies ON a circle if it lies ON the circumference

Points on a circle

• any point that lies on the circle will
satisfy the equation for the circle (below)

• any point that lies on the circle will
create a right triangle whose hypotenuse = radius

Basic equation of a circle
To determine whether a point lies on a circle,
plug the x- and y-coordinates into the equation

As abhimahna and 0akshay0 mention explicitly,
the basic equation of a circle is

\(x^2 + y^2 = r^2\)

\(x\) and \(y\) are coordinates
of a point on the circle
\(r\) = radius

Basic equation of circle \(<->\) Pythagorean theorem

Pythagorean theorem, where c = hypotenuse:
\(a^2 + b^2 = c^2\)
Basic equation of circle with center (0,0), r = radius:
\(x^2 + y^2 = r^2\)

x replaces a, y replaces b
and r, radius, replaces c, hypotenuse

Any point (x,y) on this circle therefore
satisfies the equation for THIS circle:
\(x^2 + y^2 = 10^2\)
\(x^2 + y^2 = 100\)

Scan the answers.
You're looking for x- and y-values
whose squares will sum to 100.

Plug in answers?
You can plug in until you find the right pair.

Try A. (3,4)

\(3^2 + 4^2 = r^2\)
\(r^2 = 25\)
\(r = 5\)
r must = 10

You can keep plugging in
until you get the correct x- and y- values

See the RHS of diagram.*
All points except D lie INSIDE the circle
Their radii lengths are shorter than 10

OR notice: 6-8-10 = 3-4-5 right triangle

Answer D is (8,6)
Radius = 10
8: 6: 10 = 4x-3x-5x

D's x- and y-coordinates yield lengths
that are a "Pythagorean triplet" triangle

(8, 6) makes a right triangle. See diagram.
Its hypotenuse is the radius.

Check: do D's coordinates (8,6) satisfy equation?

Answer D: (8, 6)
\(x^2 + y^2 = r^2\)
\(8^2 + 6^2 = r^2\)
\(64 + 36 = r^2\)
\(r^2 = 100\)
\(r = 10\)

That's a match.

Answer D

Hope that helps

* Look at the green triangle on the left.
The green point is (-6, 8). It, too, satisfies the equation

\((-6)^2 + 8^2 = r^2\)
\((36 + 64) = 100 = r^2\)
\(r^2 = 100\)
\(r = 10\)

For more on the equation of a circle, see Bunuel , Circle on a plane , and another site HERE


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