Tangent Line to a Circle at a Point

A tangent line is perpendicular to the radius at the point of contact. The radius from (0, 0) to (3, 4) has slope 4/3, so the tangent has slope −3/4: y − 4 = −(3/4)(x − 3), which simplifies to 3x + 4y = 25. For a circle x² + y² = r² the tangent at (x₁, y₁) is always x₁x + y₁y = r². The line y = (4/3)x is the radius itself. TI-36X Pro tip: plug the point into each answer choice: 3(3) + 4(4) = 25 works. Then confirm it is not the radius: the radius slope is 4 ÷ 3 and the tangent slope is −3 ÷ 4, and their product is −1.