Intersection of Lines and Circles Practice

Determine how many times the line y = 3 intersects the circle x^2 + y^2 = 25.

Two points. Substituting y = 3 gives x^2 + 9 = 25, so x^2 = 16, yielding x = 4 and x = -4.

Find the coordinates of the points where the line y = x + 1 meets the circle x^2 + y^2 = 13.

(2, 3) and (-3, -2). Substituting gives x^2 + (x + 1)^2 = 13, which simplifies to 2x^2 + 2x - 12 = 0, or x^2 + x - 6 = 0. Factoring gives (x + 3)(x - 2) = 0.

Use the discriminant to determine if the line y = 2x + 5 is tangent, secant, or exterior to the circle x^2 + y^2 = 5.

Tangent (one intersection point). Substituting gives x^2 + (2x + 5)^2 = 5, which simplifies to 5x^2 + 20x + 20 = 0. The discriminant is 20^2 - 4(5)(20) = 400 - 400 = 0.

Find the point of tangency between the line y = -x + 4 and the circle x^2 + y^2 = 8.

(2, 2). Substituting gives x^2 + (-x + 4)^2 = 8, leading to 2x^2 - 8x + 8 = 0, or (x - 2)^2 = 0, giving x = 2 and y = 2.

Compare the radius to the center distance to decide if the line 3x + 4y - 25 = 0 intersects the circle x^2 + y^2 = 16.

No intersection points. The center is (0, 0) and radius r = 4. The perpendicular distance from (0, 0) to 3x + 4y - 25 = 0 is |3(0) + 4(0) - 25| / sqrt(3^2 + 4^2) = 25 / 5 = 5. Since d = 5 is greater than r = 4, the line does not meet the circle.

Find the intersection points of the line y = 2x - 1 and the circle (x - 1)^2 + (y + 2)^2 = 5.

(0, -1) and (1, 1). Substituting y = 2x - 1 gives (x - 1)^2 + (2x + 1)^2 = 5, expanding to 5x^2 = 5, so x^2 = 1. Therefore, x = 1 or x = -1 (note: substituting x = -1 gives y = -3, check yields (-1 - 1)^2 + (-3 + 2)^2 = 4 + 1 = 5; points are (1, 1) and (-1, -3)).