MCQOPTIONS
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MCQOPTIONS
This section includes 24 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
The circle passing through point of intersection of the circle \[S=0\] and the line \[P=0\] is [RPET 1995] |
| A. | \[S+\lambda P=0\] |
| B. | \[S-\lambda P=0\] |
| C. | \[\lambda S+P=0\] |
| D. | \[P-\lambda S=0\] |
| E. | All of these |
| Answer» F. | |
| 2. |
The locus of the centre of a circle which touches externally the circle \[{{x}^{2}}+{{y}^{2}}-6x-6y+14=0\] and also touches the y-axis, is given by the equation [IIT 1993; DCE 2000] |
| A. | \[{{x}^{2}}-6x-10y+14=0\] |
| B. | \[{{x}^{2}}-10x-6y+14=0\] |
| C. | \[{{y}^{2}}-6x-10y+14=0\] |
| D. | \[{{y}^{2}}-10x-6y+14=0\] |
| Answer» E. | |
| 3. |
A line meets the coordinate axes in A and B. A circle is circumscribed about the triangle OAB. If m and n are the distance of the tangents to the circle at the points A and B respectively from the origin, the diameter of the circle is |
| A. | \[m(m+n)\] |
| B. | \[m+n\] |
| C. | \[n(m+n)\] |
| D. | \[\frac{1}{2}(m+n)\] |
| Answer» C. \[n(m+n)\] | |
| 4. |
A circle which passes through origin and cuts intercepts on axes a and b, the equation of circle is [RPET 1991] |
| A. | \[{{x}^{2}}+{{y}^{2}}-ax-by=0\] |
| B. | \[{{x}^{2}}+{{y}^{2}}+ax+by=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}-ax+by=0\] |
| D. | \[{{x}^{2}}+{{y}^{2}}+ax-by=0\] |
| Answer» B. \[{{x}^{2}}+{{y}^{2}}+ax+by=0\] | |
| 5. |
The equation of a diameter of circle \[{{x}^{2}}+{{y}^{2}}-6x+2y=0\] passing through origin is [RPET 1991; IIT 1989; MP PET 2002] |
| A. | \[x+3y=0\] |
| B. | \[x-3y=0\] |
| C. | \[3x+y=0\] |
| D. | \[3x-y=0\] |
| Answer» B. \[x-3y=0\] | |
| 6. |
The centre and radius of the circle \[2{{x}^{2}}+2{{y}^{2}}-x=0\] are [MP PET 1984, 87] |
| A. | \[\left( \frac{1}{4},\ 0 \right)\] and \[\frac{1}{4}\] |
| B. | \[\left( -\frac{1}{2},\ 0 \right)\] and \[\frac{1}{2}\] |
| C. | \[\left( \frac{1}{2},\ 0 \right)\] and \[\frac{1}{2}\] |
| D. | \[\left( 0,\ -\frac{1}{4} \right)\] and \[\frac{1}{4}\] |
| Answer» B. \[\left( -\frac{1}{2},\ 0 \right)\] and \[\frac{1}{2}\] | |
| 7. |
The equation of the circle passing through the point (2, 1) and touching y-axis at the origin is |
| A. | \[{{x}^{2}}+{{y}^{2}}-5x=0\] |
| B. | \[2{{x}^{2}}+2{{y}^{2}}-5x=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}+5x=0\] |
| D. | None of these |
| Answer» C. \[{{x}^{2}}+{{y}^{2}}+5x=0\] | |
| 8. |
The equation of the circle concentric with the circle \[{{x}^{2}}+{{y}^{2}}+8x+10y-7=0\] and passing through the centre of the circle \[{{x}^{2}}+{{y}^{2}}-4x-6y=0\] is |
| A. | \[{{x}^{2}}+{{y}^{2}}+8x+10y+59=0\] |
| B. | \[{{x}^{2}}+{{y}^{2}}+8x+10y-59=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}-4x-6y+87=0\] |
| D. | \[{{x}^{2}}+{{y}^{2}}-4x-6y-87=0\] |
| Answer» C. \[{{x}^{2}}+{{y}^{2}}-4x-6y+87=0\] | |
| 9. |
If a circle whose centre is (1, ?3) touches the line \[3x-4y-5=0\], then the radius of the circle is |
| A. | 2 |
| B. | 4 |
| C. | \[\frac{5}{2}\] |
| D. | \[\frac{7}{2}\] |
| Answer» B. 4 | |
| 10. |
Equation of the circle which touches the lines \[x=0,\ y=0\] and \[3x+4y=4\] is [MP PET 1991] |
| A. | \[{{x}^{2}}-4x+{{y}^{2}}+4y+4=0\] |
| B. | \[{{x}^{2}}-4x+{{y}^{2}}-4y+4=0\] |
| C. | \[{{x}^{2}}+4x+{{y}^{2}}+4y+4=0\] |
| D. | \[{{x}^{2}}+4x+{{y}^{2}}-4y+4=0\] |
| Answer» C. \[{{x}^{2}}+4x+{{y}^{2}}+4y+4=0\] | |
| 11. |
The equation of a circle which touches both axes and the line \[3x-4y+8=0\] and whose centre lies in the third quadrant is [MP PET 1986] |
| A. | \[{{x}^{2}}+{{y}^{2}}-4x+4y-4=0\] |
| B. | \[{{x}^{2}}+{{y}^{2}}-4x+4y+4=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}+4x+4y+4=0\] |
| D. | \[{{x}^{2}}+{{y}^{2}}-4x-4y-4=0\] |
| Answer» D. \[{{x}^{2}}+{{y}^{2}}-4x-4y-4=0\] | |
| 12. |
The number of circle having radius 5 and passing through the points (? 2, 0) and (4, 0) is |
| A. | One |
| B. | Two |
| C. | Four |
| D. | Infinite |
| Answer» C. Four | |
| 13. |
The equation of a circle touching the axes of coordinates and the line \[x\cos \alpha +y\sin \alpha =2\]can be |
| A. | \[{{x}^{2}}+{{y}^{2}}-2gx-2gy+{{g}^{2}}=0\], where \[g=\frac{2}{(\cos \alpha +\sin \alpha +1)}\] |
| B. | \[{{x}^{2}}+{{y}^{2}}-2gx-2gy+{{g}^{2}}=0\], where \[g=\frac{2}{(\cos \alpha +\sin \alpha -1)}\] |
| C. | \[{{x}^{2}}+{{y}^{2}}-2gx+2gy+{{g}^{2}}=0\], where \[g=\frac{2}{(\cos \alpha -\sin \alpha +1)}\] |
| D. | \[{{x}^{2}}+{{y}^{2}}-2gx+2gy+{{g}^{2}}=0\] where \[g=\frac{2}{(\cos \alpha +\sin \alpha +1)}\] |
| E. | All of these |
| Answer» F. | |
| 14. |
Let \[P({{x}_{1}},{{y}_{1}})\] and \[Q({{x}_{2}},{{y}_{2}})\]are two points such that their abscissa \[{{x}_{1}}\] and \[{{x}_{2}}\] are the roots of the equation \[{{x}^{2}}+2x-3=0\] while the ordinates \[{{y}_{1}}\] and \[{{y}_{2}}\] are the roots of the equation\[{{y}^{2}}+4y-12=0\]. The centre of the circle with PQ as diameter is [Orissa JEE 2005] |
| A. | \[(-1,-2)\] |
| B. | \[(1,\,\,2)\] |
| C. | \[(1,-2)\] |
| D. | \[(-1,2)\] |
| Answer» B. \[(1,\,\,2)\] | |
| 15. |
A variable circle passes through the fixed point (2,0) and touches the y-axis . Then the locus of its centre is [EAMCET 2002] |
| A. | A circle |
| B. | An Ellipse |
| C. | A hyperbola |
| D. | A parabola |
| Answer» E. | |
| 16. |
The equation of circle whose centre lies on \[3x-y-4=0\]and\[x+3y+2=0\]and has an area 154 square units is [DCE 2001] |
| A. | \[{{x}^{2}}+{{y}^{2}}-2x+2y-47=0\] |
| B. | \[{{x}^{2}}+{{y}^{2}}-2x+2y+47=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}+2x-2y-47=0\] |
| D. | None of these |
| Answer» B. \[{{x}^{2}}+{{y}^{2}}-2x+2y+47=0\] | |
| 17. |
The equation of the circle in the first quadrant which touches each axis at a distance 5 from the origin is [MP PET 1997] |
| A. | \[{{x}^{2}}+{{y}^{2}}+5x+5y+25=0\] |
| B. | \[{{x}^{2}}+{{y}^{2}}-10x-10y+25=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}-5x-5y+25=0\] |
| D. | \[{{x}^{2}}+{{y}^{2}}+10x+10y+25=0\] |
| Answer» C. \[{{x}^{2}}+{{y}^{2}}-5x-5y+25=0\] | |
| 18. |
If the line \[x-2y=k\]cuts off a chord of length 2 from the circle \[{{x}^{2}}+{{y}^{2}}=3\], then k = |
| A. | 0 |
| B. | \[\pm 1\] |
| C. | \[\pm \sqrt{10}\] |
| D. | None of these |
| Answer» D. None of these | |
| 19. |
The radius of the circle, having centre at (2,1) whose one of the chord is a diameter of the circle \[{{x}^{2}}+{{y}^{2}}-2x-6y+6=0\] is[IIT Screening 2004] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | \[\sqrt{3}\] |
| Answer» D. \[\sqrt{3}\] | |
| 20. |
The pole of the line \[2x+3y=4\]w.r.t circle \[{{x}^{2}}+{{y}^{2}}=64\] is [RPET 1996] |
| A. | (32, 48) |
| B. | (48, 32) |
| C. | (- 32, 48) |
| D. | (48, -32) |
| Answer» B. (48, 32) | |
| 21. |
Polar of origin (0, 0) with respect to the circle \[{{x}^{2}}+{{y}^{2}}+2\lambda x+2\mu y+c=0\] touches the circle \[{{x}^{2}}+{{y}^{2}}={{r}^{2}}\], if[RPET 1992] |
| A. | \[c=r({{\lambda }^{2}}+{{\mu }^{2}})\] |
| B. | \[r=c\,({{\lambda }^{2}}+{{\mu }^{2}})\] |
| C. | \[{{c}^{2}}={{r}^{2}}({{\lambda }^{2}}+{{\mu }^{2}})\] |
| D. | \[{{r}^{2}}={{c}^{2}}({{\lambda }^{2}}+{{\mu }^{2}})\] |
| Answer» D. \[{{r}^{2}}={{c}^{2}}({{\lambda }^{2}}+{{\mu }^{2}})\] | |
| 22. |
The length of the common chord of the circles \[{{(x-a)}^{2}}+{{(y-b)}^{2}}={{c}^{2}}\]and \[{{(x-b)}^{2}}+{{(y-a)}^{2}}={{c}^{2}}\], is |
| A. | \[\sqrt{4{{c}^{2}}-2{{(a-b)}^{2}}}\] |
| B. | \[\sqrt{4{{c}^{2}}+2{{(a-b)}^{2}}}\] |
| C. | \[\sqrt{4{{c}^{2}}-2{{(a+b)}^{2}}}\] |
| D. | \[\sqrt{4{{c}^{2}}+2{{(a+b)}^{2}}}\] |
| Answer» B. \[\sqrt{4{{c}^{2}}+2{{(a-b)}^{2}}}\] | |
| 23. |
The length of common chord of the circles \[{{(x-a)}^{2}}+{{y}^{2}}={{a}^{2}}\]and \[{{x}^{2}}+{{(y-b)}^{2}}={{b}^{2}}\]is [MP PET 1989] |
| A. | \[2\sqrt{{{a}^{2}}+{{b}^{2}}}\] |
| B. | \[\frac{ab}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\] |
| C. | \[\frac{2ab}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\] |
| D. | None of these |
| Answer» C. \[\frac{2ab}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\] | |
| 24. |
The common chord of the circle \[{{x}^{2}}+{{y}^{2}}+4x+1=0\] and \[{{x}^{2}}+{{y}^{2}}+6x+2y+3=0\] is [MP PET 1991] |
| A. | \[x+y+1=0\] |
| B. | \[5x+y+2=0\] |
| C. | \[2x+2y+5=0\] |
| D. | \[3x+y+3=0\] |
| Answer» B. \[5x+y+2=0\] | |