Problem

Decide whether the candidate line is tangent using this complete coordinate evidence: circle center \(O=(0,0)\), radius\(5\); \(T=(4,3)\) lies on the circle and on candidate line \(y=3\).

Coordinate plane with circle centered at O=(0,0) of radius 5, point T=(4,3), and candidate line y=3 through T.

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What this problem is really about

Tangency in coordinates needs two independent facts: the contact point lies on both objects, and the candidate line is perpendicular to the radius there. We’ll verify incidence, compare directions, and confirm the classification by solving for every line-circle intersection.

Answer choices

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Yes. The line is tangent at (4, 3) because it shares that point with the circle.

Yes. Two intersections make y = 3 a tangent line.

Yes. A horizontal line is perpendicular to radius OT because OT has slope 3/4.

No. Line y = 3 meets the circle at (4, 3) and (−4, 3), so it is a secant rather than a tangent.

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Decide whether the candidate line is tangent using this complete coordinate evidence: circle center \(O=(0,0)\), radius\(5\); \(T=(4,3)\) lies on the circle and on candidate line \(y=3\).

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