Question

A circle is such that it is tangent to every line of the form: $$ (x - 2)\cos\theta + (y - 2)\sin\theta = 1 $$ for all $ \theta $. Find the equation of the circle.

• A. \( x^2 + y^2 - 4x - 4y + 7 = 0 \)
• B. \( x^2 + y^2 + 4x + 4y + 7 = 0 \)
• C. \( x^2 + y^2 - 4x - 4y - 7 = 0 \)
• D. \( x^2 + y^2 + 4x + 4y - 7 = 0 \)

Hint:

This is a family of lines all at constant distance from a point. Recognize this as a condition describing a circle.

Answer 1

The Correct Option is A

The given equation: \[ (x - 2)\cos\theta + (y - 2)\sin\theta = 1 \] represents a family of lines all at distance 1 from the point \( (2, 2) \). This is the general form of all lines that are tangent to a circle centered at \( (2, 2) \) with radius 1. So: - Center of the circle = \( (2, 2) \) - Radius = 1 Equation of circle: \[ (x - 2)^2 + (y - 2)^2 = 1^2 \Rightarrow x^2 + y^2 - 4x - 4y + 4 + 4 + 4 = 1 \Rightarrow x^2 + y^2 - 4x - 4y + 7 = 0 \] \[ \boxed{x^2 + y^2 - 4x - 4y + 7 = 0} \]