JEE Main Mathematics: Intersection of a circle with a straight line or a circle Practice Questions

Correct answer: B — 2

The distance d from the center (0, 0) to the line 3x + 4y - 10 = 0 is d = (|3(0) + 4(0) - 10|)/(√(3² + 4²)) = 10/5 = 2. Since the distance d=2 is less than the radius r=3, the line intersects the circle at two distinct points.

Correct answer: C — The line is tangent to the circle

When a straight line intersects a circle at exactly one point, it means the line touches the circle at that single point, establishing that the line is tangent to the circle.

Correct answer: C — 0

The center of the circle is (0, 0) and the radius is r=5. The distance from the center to the line y=10 is |10| = 10. Since the distance (10) is greater than the radius (5), the line does not intersect the circle. Number of intersections is 0.

Correct answer: B — Distance is the radius

If a line intersects a circle at exactly one point, it means the line is tangent to the circle. The distance (d) from the center of the circle to a tangent line is exactly equal to the radius (r) of the circle.

Correct answer: B — 3

The center is (1, 2) and r=5. The minimum distance from a point (the center) to a line is the perpendicular distance. The question asks for the minimum distance between the *center* and a line passing through (4, 6). The minimum distance between the center (1, 2) and *any* line is 0 (if the line passes through the center). However, if we interpret the question as 'minimum distance between the center and a line passing through (4,6)', the minimum distance is achieved when the line passes through the center, which is 0. But standard context implies finding the distance from the center to a specific line or finding the minimum distance to a point. Rephrasing: Find the minimum distance from the center (1, 2) to a line passing through (4, 6). The minimum distance is the perpendicular distance from (1, 2) to the line passing through (4, 6) and perpendicular to it. This minimum distance is not simply related to the options. A better interpretation is finding the shortest distance from the center (1,2) to the point (4,6) itself, which is √((4-1)²+(6-2)²) = √(9+16) = 5. This distance is the minimum possible distance to a line passing through (4,6) that is perpendicular to the segment connecting the center and (4,6).

Correct answer: B — 2

The circle is centered at the origin O(0, 0) with radius R=√50 ≈ 7.07. The line is x + √3y = 10. The distance d from the origin to the line is d = (|0 + √3(0) - 10|)/(√(1² + (√3)²)) = 10/2 = 5. Since d=5 and R=√50 ≈ 7.07. Since d < R, the line intersects the circle at two distinct points.