A circle has its centre at the origin and a point P (5,0) lies on it . The point Q (6,8) lies outside the circle.

Explanation

To determine whether the statement "The point Q (6,8) lies outside the circle" is true or false, we will follow these steps:

### Step 1: Identify the center and radius of the circle
The center of the circle is at the origin (0,0) and the point P (5,0) lies on the circle. The distance from the center to point P is the radius of the circle.

**Calculation of radius:**
- The radius \( r \) is the distance from the center (0,0) to the point P (5,0).
- Using the distance formula:
\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \( (x_1, y_1) = (0, 0) \) and \( (x_2, y_2) = (5, 0) \):
\[ r = \sqrt{(5 - 0)^2 + (0 - 0)^2} = \sqrt{5^2} = 5 \]

### Step 2: Write the equation of the circle
The equation of a circle with center at (0,0) and radius \( r \) is given by:
\[ x^2 + y^2 = r^2 \]
Substituting \( r = 5 \):
\[ x^2 + y^2 = 5^2 = 25 \]

### Step 3: Determine the position of point Q (6,8)
Now we need to check if the point Q (6,8) lies inside, on, or outside the circle. We will substitute the coordinates of point Q into the circle's equation.

**Calculation:**
- Substitute \( x = 6 \) and \( y = 8 \) into the equation:
\[ 6^2 + 8^2 = 36 + 64 = 100 \]

### Step 4: Compare with the radius squared
Now we compare \( 100 \) with \( 25 \):
- Since \( 100 > 25 \), this means that the point Q (6,8) lies outside the circle.

### Conclusion
Thus, the statement "The point Q (6,8) lies outside the circle" is **True**.

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