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Find a circle from given conditions, then the length of the chord it cuts on a line

This type of question, asked 4 times: 29 Jan 2025, 2 Apr 2026 S1, 2 Apr 2026 S2, 6 Apr 2026 S1. All 4 questions below, oldest first.

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  1. Question 1

    29 Jan 2025

    139
    Let a circle CC pass through the points (4,2)(4,2) and (0(0, 2 ), and its centre lie on 3x+2y+2=03x + 2y + 2 = 0. Then the length of the chord, of the circle CC, whose midpoint is (1,2)(1,2), is:

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  2. Question 2

    2 Apr 2026 S1

    74
    Let a circle C have its center in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of C on the line x+y=1x + y = 1 is 14\sqrt{14}, then the square of the radius of C is ____\_\_\_\_ .

  3. Question 3

    2 Apr 2026 S2

    60
    Let a circle pass through the origin and its center be the point of intersection of two mutually perpendicular lines x+(k−1)y+3=0x + (k - 1)y + 3 = 0 and 2x+k2y−4=02x +k^{2}y - 4 = 0. If the line x−y+2=0x - y + 2 = 0 intersects the circle at the points A and B , then (AB)2(AB)^{2} is equal to :

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  4. Question 4

    6 Apr 2026 S1

    72
    Let the centre of the circle x2+y2+2gx+2fy+25=0x^{2} + y^{2} + 2gx + 2fy + 25 = 0 be in the first quadrant and lie on the line 2x−y=42x - y = 4. Let the area of an equilateral triangle inscribed in the circle be 27327\sqrt{3}. Then the square of the length of the chord of the circle on the line x=1x = 1 is____\_\_\_\_ .