The centre of the circle is at the origin and its radius is 10. Which of the following points lies inside the circle? 

Options

A. (6,8)
B. (0, 11)
C. (-10,0)
D. (7,7)

Answer

To determine which of the given points lies inside the circle with its center at the origin (0, 0) and a radius of 10, we can follow these steps:

### Step 1: Understand the Circle's Equation
The equation of a circle with center at the origin (0, 0) and radius \( r \) is given by:
\[ x^2 + y^2 = r^2 \]
In this case, since the radius \( r = 10 \), the equation becomes:
\[ x^2 + y^2 = 100 \]

### Step 2: Identify the Points to Check
We need to check the following points to see if they lie inside the circle:
1. (6, 8)
2. (0, 11)
3. (-10, 0)
4. (7, 7)

### Step 3: Calculate the Distance for Each Point
To determine if a point lies inside the circle, we calculate the distance from the origin using the distance formula:
\[ d = \sqrt{x^2 + y^2} \]
If \( d < 10 \), the point lies inside the circle. If \( d = 10 \), it lies on the circle, and if \( d > 10 \), it lies outside the circle.

#### Check Point (6, 8):
\[ d = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]
Since \( d = 10 \), point (6, 8) lies on the circle.

#### Check Point (0, 11):
\[ d = \sqrt{0^2 + 11^2} = \sqrt{0 + 121} = \sqrt{121} = 11 \]
Since \( d > 10 \), point (0, 11) lies outside the circle.

#### Check Point (-10, 0):
\[ d = \sqrt{(-10)^2 + 0^2} = \sqrt{100 + 0} = \sqrt{100} = 10 \]
Since \( d = 10 \), point (-10, 0) lies on the circle.

#### Check Point (7, 7):
\[ d = \sqrt{7^2 + 7^2} = \sqrt{49 + 49} = \sqrt{98} \approx 9.9 \]
Since \( d < 10 \), point (7, 7) lies inside the circle.

### Conclusion
The point that lies inside the circle is **(7, 7)**.