21). AB, is a diameter of a circle and c is any point on circumference of the circle then:
| A). the arc of \( \Large \triangle ABC \) is maximum, when it is isosceles |
| B). the area of \( \Large \triangle ABC \) is maximum, when it is isosceles |
| C). the perimeter of \( \Large \triangle ABC \) is maximum, when it is isosceles |
| D). none of the above |
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22). The number of common tangents to the circles \( \Large x^{2}+y^{2}-2x-4y+1=0 \) and \( \Large x^{2}+y^{2}-12x-16y+91=0 \) is
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23). A, B, C and D are the points of intersection with the co-ordinate axes of the lines \( \Large ax+by=ab\ and\ bx+ay=ab \) then:
| A). A, B, C, D are concyclic |
| B). A, B, C, D form a parallelogram |
| C). A, B, C, D form a rhombus |
| D). none of the above |
|
24). The gradient of the radical axis of the circles \( \Large x^{2}+y^{2}-3x-4y+5=0\ and\ 3x^{2}+3y^{2}-7x+8y+11=0 \) is
| A). \( \Large \frac{1}{3} \) |
| B). \( \Large -\frac{1}{10} \) |
| C). \( \Large -\frac{1}{2} \) |
| D). \( \Large -\frac{2}{3} \) |
|
25). The limiting point of the system of circles represented by the equation \( \Large 2\left(x^{2}+y^{2}\right)+nx+\frac{9}{2}=0 \) are
| A). \( \Large \left(\pm \frac{3}{2},\ 0\right) \) |
| B). \( \Large \left(0,\ 0\right)\ and\ \left(\frac{9}{2},\ 0\right) \) |
| C). \( \Large \left(\pm \frac{9}{2},\ 0\right) \) |
| D). \( \Large \left(\pm 3,\ 0\right) \) |
|
26). The radical centre of the circles \( \Large x^{2}+y^{2}-16x+60=0,\) \( \Large x^{2}+y^{2}-12x+27=0,\) \( \Large x^{2}+y^{2}-12y+8=0 \) is
| A). \( \Large \left(13,\ \frac{33}{4}\right) \) |
| B). \( \Large \left(\frac{33}{4},\ -13\right) \) |
| C). \( \Large \left(\frac{33}{4},\ 13\right) \) |
| D). none of these |
|
27). The circles \( \Large x^{2} + y^{2} - 10x +16 = 0 \) and \( \Large x^{2} + y^{2} = r^{2} \) intersect each other at two distinct points if:
| A). r < 2 |
| B). r > 8 |
| C). 2 < r < 8 |
| D). \( \Large 2 \le r \le 8 \) |
|
28). The centres of a set of circles, each of radius 3, lies on the circle \( \Large x^{2} + y^{2} = 25 \). The locus of any point in the set is:
| A). \( \Large 4 \le x^{2}+y^{2} \le 64 \) |
| B). \( \Large x^{2}+y^{2} \le 25 \) |
| C). \( \Large x^{2}+y^{2}\ge 25 \) |
| D). \( \Large 3 \le x^{2}+y^{2} \le 9 \) |
|
29). A variable circle passes through the fixed point A(p, q) and touches x-axis. The locus of the other end of the diameter through A is:
| A). \( \Large \left(x-p\right)^{2}=4qy \) |
| B). \( \Large \left(x-q\right)^{2}=4py \) |
| C). \( \Large \left(y-p\right)^{2}=4qx \) |
| D). \( \Large \left(y-q\right)^{2}=4py \) |
|
30). If the lines \( \Large 2x + 3y + 1 = 0 \) and \( \Large 3x - y - 4 = 0 \) lies along diameter of a circle of circumference \( \Large 10 \pi \), then the equation of the circle is:
| A). \( \Large x^{2} + y^{2} - 2x + 2y - 23 = 0 \) |
| B). \( \Large x^{2} + y^{2} - 2x - 2y - 23 = 0 \) |
| C). \( \Large x^{2} + y^{2} + 2x + 2y - 23 = 0 \) |
| D). \( \Large x^{2} + y^{2} + 2x - 2y - 23 = 0 \) |
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