The point \(P(-2,4)\) lies on a circle of radius 6 and centre \((3,5)\).
False.
Step 1: A point lies on a circle if the distance between the point and the centre of the circle is equal to the radius.
Step 2: The formula for distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 3: Here, the centre of the circle is \((3, 5)\) and the point is \((-2, 4)\).
So, \(x_1 = 3,\; y_1 = 5,\; x_2 = -2,\; y_2 = 4\).
Step 4: Substitute values:
\[ d = \sqrt{((-2) - 3)^2 + (4 - 5)^2} \]
\[ d = \sqrt{(-5)^2 + (-1)^2} \]
\[ d = \sqrt{25 + 1} \]
\[ d = \sqrt{26} \]
Step 5: Approximate value:
\( \sqrt{26} \approx 5.1 \; \text{units} \)
Step 6: Compare with the radius:
The radius = 6 units.
The distance = 5.1 units.
Step 7: Since 5.1 units is less than 6 units, the point lies inside the circle, not on it.
Final Answer: The statement is False.