Equations of Circles: Practice

The horizontal tangent has a vertical contact radius, so the center shares first coordinate negative 4 with T. Its stated height is negative 2, making the radius 5.

The alternatives give a different center or use the radius rather than its square. The recovered vertical radius is perpendicular to the given line at T.

Substitute and complete the remaining square.

Exactly one real first coordinate occurs when R is 8, forcing x to be 3 and y to be 1. Smaller listed values give no real root; 16 gives two. The contact-radius slope is negative 1 and the line slope is 1, confirming tangency.

Substituting the height gives

The endpoints are (-1,2) and (7,2). Averaging the first coordinates gives

Their common height remains 2. The point (3,-1) is the center, while the other choices are not halfway between the endpoints.

Substitute the line equation and factor.

The remaining first coordinate is negative 6/5; the line gives height 17/5. Its squared coordinate sum is

The positive 6/5 choice satisfies the line but has squared sum 85/25, not 13. The point (-2,5) gives 29, and (2,3) fails the line.

Substitute the fixed first coordinate.

One real height occurs at q equal to 9, forcing y to be negative 1. The contact radius is horizontal, perpendicular to the vertical line. The value 3 gives no intersection; (2,-1) is the center rather than a line point; zero is not a positive squared radius.

Complete the squares to recover the center.

The x-axis is 4 units above the center. Tangency therefore requires squared radius 16.

This gives K equal to 9, with contact (-3,0) and a vertical radius. At 25 the graph is one point off the axis; at 16 the radius is 3 and the axis misses; at 41 there are no real points.

The segment lies on the line with equation

Substitution into the circle gives

The points (1,2) and (2,1) both lie between the stated endpoints and have squared coordinate sum 5. The endpoints have squared sum 9. The negative-height pairs fail the segment line.

The squared distance from C to the given foot is

This is less than 20. The perpendicular foot lies inside the circle, so the line crosses the circle on both sides of H. The numerical comparison and squared-distance values in the other choices are false.

Complete the squares.

After substituting the line height, two distinct first coordinates require a positive right side in

The vertical gap must have magnitude less than 4, giving heights strictly between negative 7 and 1. At the endpoints there is one contact; beyond them there is none. The last interval uses heights measured from zero rather than the center height negative 3.

The new center is (1,-3) and the radius remains 2, giving

At first coordinate zero,

There are two distinct real heights. The fixed line is a secant and does not pass through the new center, whose first coordinate is 1.

The sum of squares can be zero only at (8,-2). That point has a different height from the line. Substitution confirms this.

There is no real root. The zero-right-side graph is a point, so positive-radius circle tangency rules do not apply.

A center of height 1 must have first coordinate 3 units from the vertical tangent x equals negative 2. The possibilities are 1 and negative 5.

For center (1,1), the squared distance to (1,4) is

For center (-5,1), it is

Only (1,1) works. The contact (-2,1) has a horizontal radius, perpendicular to the vertical tangent. The radius-9 option violates the radius condition, and the center-height-4 option violates the center line.