JEE Mathematics Online Test Circle Set C

1.

2. Statement-1 : The product of slopes of all the common tangents of circles S₁ = x² + y² – a² = 0 and S₂ = (x – 2a)² + (y – 2a)² – 4a² = 0 is 1. Statement-2 : Slope of line joining centres of S₁ = x² + y² – a² = 0 and S² = (x – 2a)² + (y – 2a)² – 4a² = 0 is 1. Direct common tangents make equal angles with the line joining centres of the circles.

3. Statement-1 : Number of common tangents of x² + y² – 2x – 4y – 95 = 0 and x² + y² – 6x – 8y + 16 = 0 is zero.
Statement 2 : If C₁C₂ < | r₁ – r₂ |, then there will be no common tangent. (where C₁, C₂ are the centre and r₁, r₂ are radii of circles).

4. A circle x² + y² = 1 cuts the line x + y = k and makes the chord with length l. The value of k is – 1. √6/2 2. -√6/2 3. √3/2 4. None of the options

5. Consider the circle x² + y² – 10x – 6y + 30 = 0. Let O be the centre of the circle and tangent at A (7, 3) and B (5, 1) meet at C. Let S = 0 represents family of circles passings through A and B, then – (1) Area of quadrilateral OACB = 4
(2) The smallest possible circle of the family of circles S = 0 is x² + y² – 12x – 4y + 38 = 0
(3) The coordinates of point C are (7, 1)
(4) The radical axis for the family of circles S = 0 is x + y = 10

6. The equations of the tangents drawn from the origin to the
circle x² + y² – 2rx – 2hy + h² = 0, are (1) x = 0
(2) y = 0
(3) (h² – r²)x – 2rhy = 0
(4) (h² – r²)x + 2rhy = 0

7. The pole of the straight line 9x + y – 28 = 0 with respect to the circle 2x² + 2y² – 3x + 5y – 7 = 0, is

8. The area of the triangle formed by the tangents from an external point (h, k) to the circle x² + y² = a² and the chord of contact, is -

9. Length of the tangent drawn from point (1, 5) to the circle
2x² + 2y² = 3 is -

10. If the tangent to a circle x² + y² = 5 at point (1, –2) touches the circle x² +y² – 8x +6y+ 20 = 0, then its point of contact is-

11. The equation of a normal to the circle
x² + y² + 6x + 8y + 1 = 0 passing through (0, 0) is -

12. If a chord of the circle x² + y² = 8 makes equal intercepts of length a on the coordinate axes, then-

13. The locus of the point, the chord of contact of tangents from which to the circle x² + y² = a² subtends a right angle at the centre is a circle of radius -

14. The equation of the radical axis of two circles
x² + y² – x + 1 = 0 and 3(x² + y²) + y – 1 = 0 is -

15. The radical centre of the three circles
x² + y² = a²,(x – c)² + y² = a² and x² + (y – b)² = a² is -

16. The pole of the line x/a + y/b = 1 with respect to circle x² + y² = c2 is

17. Equation of polar of point (4, 4) with respect to circle (x –1)² + (y – 2)² = 1 is

18. For what value of k the circles x² + y² + 5x + 3y + 7 = 0 and x² + y² – 8x + 6y + k = 0 cuts orthogonally ?

19. If the circle x² + y² + 4x + 22y + c = 0 bisects the circumference of the circle x² + y² – 2x + 8y – d = 0, then c + d =

20. The equation of the circle which passes through points of intersection of circle x² + y² + 4x – 5y + 3 = 0 and x² + y² + 2x + 3y – 3 = 0 and point (–3, 2) is -

21. If lines y = x + 3 cuts the circle x² + y² = a2 in two points A and B, then equation of circle with AB as diameter is -

22. The equation of chord of the circle x² + y² = 8x bisected at the point (4, 3) is

23. The two circles x² + y² = ax and x² + y² = c² (with c > 0) touch each other if -

24. The locus of the centre of the circle which touches externally the circle x² + y² – 6x – 6y + 14 = 0 and also touches the y-axis, is -

25. Distance between the chord of contact with respect to point (0, 0) and (g, f) of circle x² + y² + 2gx + 2fy + c = 0 is