Question

A circle centered at the origin contains the point (0, –9). Does (8, ) also lie on the circle? Explain.
No, the distance from the center to the point
(8, ) is not the same as the radius.
No, the radius of 10 units is different from the distance from the center to the point
(8, ).
Yes, the distance from the origin to the point
(8, ) is 9 units.
Yes, the distance from the point (0, –9) to the point (8, ) is 9 units.

Answer 1

Answer:

Yes, the distance from the origin to the point $(8,\sqrt{17})$ is $9$ units.

Step-by-step explanation:

The question is incomplete. It should say ''Does $(8,\sqrt{17})$ also lie on the circle? Explain''.

The equation of a circle centered at the origin is the following :

$x^{2}+y^{2}=R^{2}$ (I)

Where ''$R$'' is a real number and the radius of the circle.

We know that the circle contains the point $(0,-9)$. Therefore, we can replace this information in the equation (I) to obtain the radius of the circle ⇒

$x^{2}+y^{2}=R^{2}$ ⇒

$(0)^{2}+(-9)^{2}=R^{2}$ ⇒

$R^{2}=81$ (II)

And finally the radius will be $R=\sqrt{81}$ ⇒ $R=9$ units

If we replace (II) in (I) we obtain :

$x^{2}+y^{2}=81$ (III)

(III) is the complete equation of the circle centered at the origin that contains the point $(0,-9)$

To see if the point $(8,\sqrt{17})$ lie on the circle we only need to replace the coordinates of the point in the equation (III) :

$x^{2}+y^{2}=81$

$(8)^{2}+(\sqrt{17})^{2}=81$

$64+17=81$

$81=81$

The point $(8,\sqrt{17})$ satisfies the equation (III). Therefore, it lies on the circle which equation is (III).

Now, to pick an option we need to remember the definition of a circle (centered at the origin) respect to the equation (I) :

$x^{2}+y^{2}=R^{2}$ (I)

The points $(x,y)$ that verifies the equation (I) are located a distance ''$R$'' from the origin.

The correct option is :

Yes, the distance from the origin to the point $(8,\sqrt{17})$ is $9$ units.

Answer 2

Yes, the distance from the origin to the point $(8, \sqrt{17} )$ is 9 units.

Calculations and Parameters:

Given the equation of the circle centered at the origin is $x^2 + y^2= R^2$.......(i)

Where R is a real number.

Knowing that the circle contains a point (0, -9), we would replace that in eq (i) to get the circle radius.

We would get $R^2= 81$........ (ii)

Hence, the radius of R would be 9 units.

If we replace (II) in (I) we obtain :

$x^2 + y^2 = 81$....... (iii)

Therefore, eq (iii) contains the complete equation of the circle centered at the origin that contains the point (0, -9)

The point $(8, \sqrt{17} )$ satisfies the equation (III).

Therefore, it lies on the circle which equation is (III).

Recall eq (i) where $x^2 + y^2= R^2$

Hence, the distance from the origin to the point $(8, \sqrt{17} )$ is 9 units.

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