Question

which of the following represents a circle with its center at the origin and a radius of 10?
a. $x^{2}+y^{2}=10$
b. $x^{2}-y^{2}=100$
c. $x^{2}+y^{2}=100$
d. $(x + 10)^{2}+(y + 10)^{2}=100$
a restaurant is located at the point $(1,2)$, and they have a delivery radius of 5 miles. a house located at the point $(5,6)$ requests a delivery. is the house within the delivery radius?
a. no, because the house does not exist
b. yes, because the point lies inside the circle
c. yes, because the point lies on the edge of the circle
d. no, because the point lies outside the circle

Explanation:

Step1: Recall the standard equation of a circle

The standard equation of a circle with center \((h,k)\) and radius \(r\) is \((x - h)^2+(y - k)^2=r^2\). When the center is at the origin \((0,0)\) and \(r = 10\), the equation is \(x^{2}+y^{2}=10^{2}=100\).

Step2: Calculate the distance from the center of the circle (origin \((0,0)\)) to the point \((5,6)\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1 = 0,y_1=0,x_2 = 5,y_2 = 6\), so \(d=\sqrt{(5 - 0)^2+(6 - 0)^2}=\sqrt{25 + 36}=\sqrt{61}\approx7.81\).

Step3: Compare the distance with the radius

Since the radius \(r = 10\) and \(d=\sqrt{61}\approx7.81<10\), the point \((5,6)\) lies inside the circle.

Answer:

For the first question: c. \(x^{2}+y^{2}=100\)
For the second question: b. Yes, because the point lies inside the circle