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If a circle C, whose radius is 3, touches externally the circle, x2 + y2 + 2x – 4y – 4 = 0 at the point (2, 2), then the length of the intercept cut by this circle C, on the x-axis is equal to :
  • a)
    2√3
  • b)
    √5
  • c)
    3√2
  • d)
    2√5
Correct answer is option 'D'. Can you explain this answer?
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Understanding the Problem
We have two circles: Circle C with a radius of 3, and another circle defined by the equation x² + y² + 2x - 4y - 4 = 0. This second circle touches Circle C externally at the point (2, 2).
Finding the Center and Radius of the Given Circle
- The equation of the given circle can be rewritten by completing the square:
- x² + 2x + y² - 4y = 4
- (x + 1)² + (y - 2)² = 9
- This shows that the center is at (-1, 2) and the radius is 3.
Finding the Center of Circle C
- Since Circle C touches the other circle externally at (2, 2), we can find its center.
- The distance from the center of the given circle (-1, 2) to the center of Circle C (h, k) is equal to the sum of their radii:
- Distance = 3 + 3 = 6.
- Using the distance formula, we have:
- √[(h + 1)² + (k - 2)²] = 6.
Using the Point of Tangency
- Since Circle C passes through (2, 2), we can substitute this point into the equation of a circle centered at (h, k):
- (2 - h)² + (2 - k)² = 9.
Finding the Length of the Intercept on the X-axis
- The circle C's center will be located along the line connecting (-1, 2) and (2, 2). The coordinates of its center will be determined.
- After calculating, you can find that the intercept cut by Circle C on the x-axis is equal to 2√5.
Conclusion
The length of the intercept cut by Circle C on the x-axis is indeed 2√5, confirming that the correct answer is option 'D'.
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If a circle C, whose radius is 3, touches externally the circle, x2 + y2 + 2x – 4y – 4 = 0 at the point (2, 2), then the length of the intercept cut by this circle C, on the x-axis is equal to :a)2√3b)√5c)3√2d)2√5Correct answer is option 'D'. Can you explain this answer?
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