(Answered)-cirlcle level -1 1. The equation ax2 + 2hxy + by2 + 2gx + 2fy + c - (2025 Updated Original AI-Free Solution

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cirlcle

level -1

1. The equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a circle, the condition will be (A) a = b and c = 0 (B) f = g and h = 0 (C) a = b and h = 0 (D) f = g and c = 0

2. The equation x2 + y2 + 2gx + 2fy + c = 0 represents a real circle if (A) g2 + f2 ? c < 0 (B) g2 + f2 ? c ? 0 (C) always (D) none of these

3. Equation of a circle with centre (4,3) touching the circle x2 + y2 = 1 is (A) x2 +y2 ? 8x ? 6y ? 9 = 0 (B) x2 + y2? 8x ? 6y + 11 = 0 (C) x2 + y2 ? 8x ?6y ? 11 = 0 (D) x2 + y2 ? 8x ? 6y + 9 = 0

4. A square is inscribed in the circle x2 + y2 ?2x + 4y + 3 = 0. Its sides are parallel to the axes. Then the one vertex of the square is (A) (1 + 2 , -2) (B) (1 - 2 , -2) (C) (1, -2 + 2 ) (D) none of these

5. The number of common real tangents that can be drawn to the circle x2 + y2 ?2x ?2y = 0 and x2 + y2 ?8x ?8y + 14 = 0 is____________________________

6. The lines 3x ?4y + 4 = 0 and 6x ?8y ?7 = 0 are tangents to the same circle. The radius of the circle is____________________________________________

7. The straight line y = mx + c cuts the circle x2+y2 = a2 at real points if (A) c )m1(a 22 ? ? (B) c )m1(a 22 ? ?

8. A line is drawn through a fixed point P (?,?) to cut the circle x2+y2 = r2 at A and B. Then PA.PB is equal to (A) (?+?)2-r2 (B) ?2+?2-r2 (C) (?-?)2+r2 (D) None of these

9. The locus of the centre of a circle of radius 2 units which rolls on the outside of the circle x2 + y2 +3x ?6y ?9 = 0 is ???????????????????????????.

10. The values of a and b for which the two circles : x2 + y2 + 2(1 ?a)x + 2(1 + b)y + (2 ?c) = 0 and x2 + y2 + 2(1 + a)x + 2(1 ?b)y + (2 + c) = 0 cut orthogonally are ?????.???????????.

11. A circle of radius 2 lies in the first quadrant and touches both the axes of co-ordinates. Then the equation of the circle with centre (6, 5) and touching the above circle externally is (A) (x ?6)2 + (y ?5)2 = 4 (B) (x ?6)2 + (y ?5)2 = 9 (C) (x ?6)2 + (y ?5)2 = 36 (D) none of these

12. Two circles x2 + y2 ? 2x ? 3 = 0 and x2 + y2 ? 4x ? 6y ? 8 = 0 are such that (A) they touch each other (B) they intersect each other (C) one lies inside the other (D) each lies outside the other
13. The least distance of point (10, 7) from the circle x2 + y2 ? 4x ? 2y ? 20 = 0 is (A) 10 (B) 15 (C) 5 (D) none of these
14. The number of common tangents to the circles x2 + y2 ? x = 0 and x2 + y2 + x=0 is (A) 2 (B) 1 (C) 4 (D) 3
15. The radius of the circle passing through the point (2, 6) two of whose diameters are x + y = 6 and x + 2y = 4 is (A) 10 (B) 2 5 (C) 6 (D) 4
16. The intercept on the line y = x by the circle x2 + y2 ? 2x = 0 is AB. The equation of the circle with AB as diameter is (A) x2 + y2 + x + y = 0 (B) x2 + y2 = x + y (C) x2 + y2 ? 3x + y = 0 (D) none of these
17. Equation of tangent to the circle x2 + y2 + 2x ? 2y + 1 = 0 at (0, 1) (A) x = 0 (B) y = 0 (C) xy = 0 (D) none of these
18. The equation x2 + y2 ? 2x + 4y + 5 = 0 represents (A) a point (B) a pair of straight lines (C) a circle (D) none of these
19. The equation of the chord of the circle x2 + y2? 4x = 0 which is bisected at the point (1, 1) is (A) x + y = 2 (B) 3x ? y = 2 (C) x ? 2y + 1 = 0 (D) x ? y = 0
20. The line ?x + ?y = 1 is a normal to the circle 2x2 + 2y2 ? 5x + 6y ?1 = 0 if (A) 5? ? 6? = 4 (B) 4 + 5? = 6? (C) 4 + 6? = 5? (D) none of these 21. The locus of the point (3h+2, k), where (h, k) lies on the circle x2+y2 = 1 is (A) a hyperbola (B) a circle (C) a parabola (D) an ellipse