Consider a family of circles which are passing through the point (-1, 1) and are tangent to the x -axis. If (h, k) are the coordinates of the centre o - Mathematics MCQ AIEEE 2007 2026 | ExamTest.live

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Consider a family of circles which are passing through the point (- 1, 1) and are tangent to the x -axis. If (h, k) are the coordinates of the centre of the circles, then the set of values of k is given by the interval:

Solution & Step-by-step Explanation

Since the circle is tangent to the x -axis, the radius r is equal to the absolute value of the y -coordinate of the centre, i.e., r = ∣ k ∣ .The equation of the circle is (x - h)^{2} + (y - k)^{2} = k^{2} .The circle passes through (- 1, 1) : (- 1 - h)^{2} + (1 - k)^{2} = k^{2} (1 + h)^{2} + 1 - 2 k + k^{2} = k^{2} (1 + h)^{2} + 1 - 2 k = 0 (1 + h)^{2} = 2 k - 1 Since (1 + h)^{2} \geq 0 for all real h : 2 k - 1 \geq 0 k \geq 1/2

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Consider a family of circles which are passing through the point (- 1, 1) and are tangent to the x -axis. If (h, k) are the coordinates of the centre of the circles, then the set of values of k is given by the interval:

0 < k < 1/2

k \geq 1/2

- 1/2 \leq k \leq 1/2

k \leq 1/2

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