31). If the circles \( \Large x^{2}+y^{2}+2x+2ky+6=0 \) and \( \Large x^{2}+y^{2}+2ky+k=0 \) intersect orthogomally then k is:
A). \( \Large 2\ or\ -\frac{3}{2} \) |
B). \( \Large -2\ or\ \frac{3}{2} \) |
C). \( \Large 2\ or\ \frac{3}{2} \) |
D). \( \Large -2\ or\ -\frac{3}{2} \) |
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32). If a > 2b > 0, then the positive value of m for which \( \Large y=mx-b\sqrt{1+m^{2}} \) is common tangent to \( \Large x^{2}+y^{2}=b^{2}\ and\ \left(x-a\right)^{2}+y^{2}=b^{2} \) is:
A). \( \Large \frac{2b}{\sqrt{a^{2}-4b^{2}}} \) |
B). \( \Large \frac{\sqrt{a^{2}-4b^{2}}}{2b} \) |
C). \( \Large \frac{2b}{a-2b} \) |
D). \( \Large \frac{b}{a-2b} \) |
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33). If \( \Large \left(a\ \cos \theta _{i},\ a\ \sin \theta _{i}\right)i = 1,\ 2,\ 3 \) represent the vertices of an equilateral triangle inscribed in a circle then:
A). \( \Large \cos \theta _{1}+\cos \theta _{2}+\cos \theta _{3} = 0 \) |
B). \( \Large \sec \theta _{1}+\sec \theta _{2}+\sec \theta _{3} = 0 \) |
C). \( \Large \tan \theta _{1}+\tan \theta _{2}+\tan \theta _{3} = 0 \) |
D). \( \Large \cot \theta _{1}+\cot \theta _{2}+\cot \theta _{3} = 0 \) |
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34). The distinct point \( \Large A \left(0,\ 0\right),\ B \left(0,\ 1\right),\ C \left(1,\ 0\right) and D \left(2a,\ 3a\right) \) are concylic then:
A). 'a' can attain only rational values |
B). 'a' is irrational |
C). cannot be concyclic for any 'a' |
D). none of the above . |
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35). The number of rational point (S) (a point \( \Large \left(a,\ b \right) \) is called rational, if a and b both are rational number) on the circumference of a circle having centre \( \Large \left( \pi ,\ e\right) \) is:
A). at most one |
B). at least two |
C). exactly |
D). infinite |
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36). If the chord of contact of tangent drawn from a point on the circle \( \Large x^{2}+y^{2}=a^{2} \) to the circle \( \Large x^{2}+y^{2}=b^{2} \) touches the circle \( \Large x^{2}+y^{2}=c^{2} \) then a, b, c are in
A). AP |
B). GP |
C). HP |
D). none of these |
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37). The condition that the chord \( \Large x \cos \alpha +y \sin \alpha - p = 0 \) of \( \Large x^{2}+y^{2}-a^{2}=0 \) may subtend a right angle at the centre of circle is:
A). \( \Large a^{2} = 2p^{2} \) |
B). \( \Large p^{2} = 2a^{2} \) |
C). \( \Large a = 2p \) |
D). \( \Large p = 2a \) |
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38). The equation of the circle passing through \( \Large \left(1,\ 1\right) \) and the points of intersection of \( \Large x^{2}+y^{2}+13x-3y=0 \) and \( \Large 2x^{2}+2y^{2}+4x-7y-25=0 \) is:
A). \( \Large 4x^{2}+4y^{2}-30x-10y=25 \) |
B). \( \Large 4x^{2}+4y^{2}+30x-13y-25=0 \) |
C). \( \Large 4x^{2}+4y^{2}-17x-10y+25=0 \) |
D). none of these |
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39). A isoscles triangle is inscribed in the circle \( \Large x^{2}+y^{2}-6x-8y=0 \) with vertex at the origin and one of the equal side along the axis of x. Equation of the other side through the origin is:
A). \( \Large 7x-24y=0 \) |
B). \( \Large 24x-7y=0 \) |
C). \( \Large 7x+24y=0 \) |
D). \( \Large 24x+7y=0 \) |
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40). If the lines \( \Large 3x-4y-7=0\ and\ 2x-3y-5=0 \) are two diameters of a circle of area \( \Large 49 \pi \) sq unit, the equation of the circle is:
A). \( \Large x^{2}+y^{2}+2x-2y-62=0 \) |
B). \( \Large x^{2}+y^{2}-2x+2y-62=0 \) |
C). \( \Large x^{2}+y^{2}-2x+2y-47=0 \) |
D). \( \Large x^{2}+y^{2}+2x-2y-47=0 \) |
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