15. Let r be a positive real number. The equation for a circle of radius r whose center is the origin is x² + y² = r². dy (a) Use implicit differentiation to determine dx (b) Let (a, b) be a point on the circle with a # 0 and b # 0. Determine the slope of the line tangent to the circle at the point (a, b). (c) Prove that the radius of the circle to the point (a, b) is perpendicular to the line tangent to the circle at the point (a, b). Hint: Two lines (neither of which is horizontal) are perpendicular if and only if the products of their slopes is equal to –1.

Question

Writing and Proofs:

15. Let r be a positive real number. The equation for a circle of radius r whose
center is the origin is x² + y² = r².
dy
(a) Use implicit differentiation to determine
dx
(b) Let (a, b) be a point on the circle with a # 0 and b # 0. Determine
the slope of the line tangent to the circle at the point (a, b).
(c) Prove that the radius of the circle to the point (a, b) is perpendicular to
the line tangent to the circle at the point (a, b). Hint: Two lines (neither
of which is horizontal) are perpendicular if and only if the products of
their slopes is equal to –1.
Expand
Transcribed Image Text

15. Let r be a positive real number. The equation for a circle of radius r whose center is the origin is x² + y² = r². dy (a) Use implicit differentiation to determine dx (b) Let (a, b) be a point on the circle with a # 0 and b # 0. Determine the slope of the line tangent to the circle at the point (a, b). (c) Prove that the radius of the circle to the point (a, b) is perpendicular to the line tangent to the circle at the point (a, b). Hint: Two lines (neither of which is horizontal) are perpendicular if and only if the products of their slopes is equal to –1.

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