Circle

Concept: Coordinate Geometry - Intersection of a Line and a Circle.

Step 1: Find the center and radius of the given circle. The equation of the circle is . We can rewrite this in standard form by completing the square: .
This simplifies to . Thus, the center is and the radius is .

Step 2: State the condition for a line to intersect a circle at two distinct points. For a line to intersect a circle at two distinct points, the perpendicular distance ( ) from the center of the circle to the line must be strictly less than the radius of the circle ( ).

Step 3: Calculate the perpendicular distance from the center to the line. The line equation is . The perpendicular distance from the center to this line is given by the formula . Substituting our values gives: .

Step 4: Apply the condition and solve the inequality for . Setting , we get . Multiplying both sides by yields . Removing the absolute value gives the compound inequality . Subtracting 3 from all parts gives .

Step 5: Count the integer values of . The problem states that (m is an integer). The integers strictly between -8 and 2 are: -7, -6, -5, -4, -3, -2, -1, 0, 1. Counting these yields a total of 9 possible values. $ $

Step 1: Use the formula for the area of a circle
The area of a circle is given by the formula: where: - is the radius of the circle, - .
Step 2: Substitute the given radius
Given: - Radius . Substitute into the formula:
Answer:
Therefore, the area of the circle is . So, the correct answer is option (1).