For which value(s) of the constant k is the circle x² + (y − k)² = 16 tangent to the line y = 3?

-1, 7

Step-by-step explanation:

Equation of the circle:

x² + (y − k)² = 16

When the circle intersects y = 3:

x² + (3 − k)² = 16

x² + 9 − 6k + k² = 16

x² = 7 + 6k − k²

x = ±√(7 + 6k − k²)

For the circle to be tangent to the line, it can only intersect at one point.  If x has only one value, then:

√(7 + 6k − k²) = -√(7 + 6k − k²)

2√(7 + 6k − k²) = 0

7 + 6k − k² = 0

k² − 6k − 7 = 0

(k − 7) (k + 1) = 0

k = -1, 7

The two values of k are -1 and 7.

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