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Find the equation of the circle that touches the line x + y = 5 at the point N(-2, 7) and cuts the circle x² + y² + 4x - 6y + 9 = 0 orthogonally.

1. x² + y² + 7x - 11y + 38 = 0
2. x² + y² = 53
3. x² + y² + x - y - 44 = 0
4. x² + y² - x + y - 62 = 0

Correct answer: x² + y² + 7x - 11y + 38 = 0

Solution

Writing the tangent family (x+2)²+(y-7)²+lambda(x+y-5)=0 and applying the orthogonality condition with the given circle yields lambda = 3, giving x² + y² + 7x - 11y + 38 = 0.

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