A circle always passes through the fixed points (a, 0) and (a, 0).Q.What is the equation to circle which touches both the axes and has centre on the line x + y = 4?a)x2 + y2 4x + 4y + 4 = 0b)x2 + y2 4x 4y + 4 = 0c)x2 + y2 + 4x 4y 4 = 0d)x2 + y2 + 4x + 4y 4 = 0Correct answer is option 'B'. Can you explain this answer?

Question

A circle always passes through the fixed points (a, 0) and (a, 0).Q.What is the equation to circle which touches both the axes and has centre on the line x + y = 4?a)x2 + y2  4x + 4y + 4 = 0b)x2 + y2  4x  4y + 4 = 0c)x2 + y2 + 4x  4y  4 = 0d)x2 + y2 + 4x + 4y  4 = 0Correct answer is option 'B'. Can you explain this answer?

Answer

The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Given information:
- The circle passes through the fixed points (a, 0) and (-a, 0).
- The center of the circle lies on the line x + y = 4.
To find the equation of the circle that satisfies these conditions, we can follow these steps:
1. Find the center of the circle:
- The center of the circle lies on the line x + y = 4.
- Since the circle also passes through (a, 0) and (-a, 0), the center must lie on the perpendicular bisector of the line segment joining these two points.
- The perpendicular bisector of the line segment joining (a, 0) and (-a, 0) is the y-axis (x = 0).
- Therefore, the center of the circle is (0, 2).
2. Find the radius of the circle:
- The radius of the circle is the distance between the center (0, 2) and one of the fixed points.
- Let's use the point (a, 0) to find the radius.
- The distance between (0, 2) and (a, 0) is given by the formula: sqrt((x2 - x1)^2 + (y2 - y1)^2)
- Substituting the values, the radius is sqrt((a - 0)^2 + (0 - 2)^2) = sqrt(a^2 + 4)
3. Substitute the center and radius into the equation of a circle:
- The center of the circle is (0, 2) and the radius is sqrt(a^2 + 4).
- Substituting these values into the equation of a circle, we get:
(x - 0)^2 + (y - 2)^2 = (sqrt(a^2 + 4))^2
Simplifying, we have:
x^2 + y^2 - 4y + 4 = a^2 + 4
x^2 + y^2 - 4y - a^2 = 0
4. Rearrange the equation to match the given options:
- Comparing the equation obtained above with the given options, we can see that option B matches the equation:
x^2 + y^2 - 4y - a^2 = 0
Therefore, the answer is option B: x^2 + y^2 - 4x - 4y + 4 = 0.

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