-
question_answer1)
The equation of \[2{{x}^{2}}+3{{y}^{2}}-8x-18y+35=k\] represents [IIT 1994]
A)
No locus if \[k>0\] done
clear
B)
An ellipse, if \[k<0\] done
clear
C)
A point if,\[k=0\] done
clear
D)
A hyperbola, if \[k>0\] done
clear
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question_answer2)
The number of points of intersection of the two curves\[y=2\sin x\] and \[y=5{{x}^{2}}+2x+3\] is [IIT 1994]
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
\[\infty \] done
clear
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question_answer3)
If the chord joining the points \[(at_{1}^{2},\ 2a{{t}_{1}})\] and \[(at_{2}^{2},\ 2a{{t}_{2}})\] of the parabola \[{{y}^{2}}=4ax\] passes through the focus of the parabola, then [MP PET 1993]
A)
\[{{t}_{1}}{{t}_{2}}=-1\] done
clear
B)
\[{{t}_{1}}{{t}_{2}}=1\] done
clear
C)
\[{{t}_{1}}+{{t}_{2}}=-1\] done
clear
D)
\[{{t}_{1}}-{{t}_{2}}=1\] done
clear
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question_answer4)
The locus of the midpoint of the line segment joining the focus to a moving point on the parabola \[{{y}^{2}}=4ax\] is another parabola with the directrix [IIT Screening 2002]
A)
\[x=-a\] done
clear
B)
\[x=-\frac{a}{2}\] done
clear
C)
\[x=0\] done
clear
D)
\[x=\frac{a}{2}\] done
clear
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question_answer5)
On the parabola \[y={{x}^{2}}\], the point least distance from the straight line \[y=2x-4\] is [AMU 2001]
A)
(1, 1) done
clear
B)
(1, 0) done
clear
C)
(1, ?1) done
clear
D)
(0, 0) done
clear
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question_answer6)
The length of the latus-rectum of the parabola whose focus is \[\left( \frac{{{u}^{2}}}{2g}\sin 2\alpha ,\ -\frac{{{u}^{2}}}{2g}\cos 2\alpha \right)\] and directrix is \[y=\frac{{{u}^{2}}}{2g}\], is
A)
\[\frac{{{u}^{2}}}{g}{{\cos }^{2}}\alpha \] done
clear
B)
\[\frac{{{u}^{2}}}{g}\cos 2\alpha \] done
clear
C)
\[\frac{2{{u}^{2}}}{g}{{\cos }^{2}}2\alpha \] done
clear
D)
\[\frac{2{{u}^{2}}}{g}{{\cos }^{2}}\alpha \] done
clear
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question_answer7)
The line \[x-1=0\] is the directrix of the parabola \[{{y}^{2}}-kx+8=0\]. Then one of the values of k is [IIT Screening 2000]
A)
\[\frac{1}{8}\] done
clear
B)
8 done
clear
C)
4 done
clear
D)
\[\frac{1}{4}\] done
clear
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question_answer8)
The centre of the circle passing through the point (0, 1) and touching the curve \[y={{x}^{2}}\]at (2, 4) is [IIT 1983]
A)
\[\left( \frac{-16}{5},\ \frac{27}{10} \right)\] done
clear
B)
\[\left( \frac{-16}{7},\ \frac{5}{10} \right)\] done
clear
C)
\[\left( \frac{-16}{5},\ \frac{53}{10} \right)\] done
clear
D)
None of these done
clear
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question_answer9)
Consider a circle with its centre lying on the focus of the parabola \[{{y}^{2}}=2px\] such that it touches the directrix of the parabola. Then, a point of intersection of the circle and the parabola is [IIT 1995]
A)
\[\left( \frac{p}{2},\ p \right)\] done
clear
B)
\[\left( \frac{p}{2},\ -p \right)\] done
clear
C)
\[\left( \frac{-p}{2},\ p \right)\] done
clear
D)
\[\left( \frac{-p}{2},\ -p \right)\] done
clear
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question_answer10)
Which one of the following curves cuts the parabola \[{{y}^{2}}=4ax\] at right angles [IIT 1994]
A)
\[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] done
clear
B)
\[y={{e}^{-x/2a}}\] done
clear
C)
\[y=ax\] done
clear
D)
\[{{x}^{2}}=4ay\] done
clear
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question_answer11)
The angle of intersection of the curves \[{{y}^{2}}=2x/\pi \] and \[y=\sin x\], is [Roorkee 1998]
A)
\[{{\cot }^{-1}}(-1/\pi )\] done
clear
B)
\[{{\cot }^{-1}}\pi \] done
clear
C)
\[{{\cot }^{-1}}(-\pi )\] done
clear
D)
\[{{\cot }^{-1}}(1/\pi )\] done
clear
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question_answer12)
The equation of the common tangent to the curves \[{{y}^{2}}=8x\] and \[xy=-1\] is [IIT Screening 2002]
A)
\[3y=9x+2\] done
clear
B)
\[y=2x+1\] done
clear
C)
\[2y=x+8\] done
clear
D)
\[y=x+2\] done
clear
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question_answer13)
The equation of the parabola whose focus is the point (0, 0) and the tangent at the vertex is \[x-y+1=0\]is [Orissa JEE 2002]
A)
\[{{x}^{2}}+{{y}^{2}}-2xy-4x+4y-4=0\] done
clear
B)
\[{{x}^{2}}+{{y}^{2}}-2xy+4x-4y-4=0\] done
clear
C)
\[{{x}^{2}}+{{y}^{2}}+2xy-4x+4y-4=0\] done
clear
D)
\[{{x}^{2}}+{{y}^{2}}+2xy-4x-4y+4=0\] done
clear
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question_answer14)
If \[a\ne 0\] and the line \[2bx+3cy+4d=0\] passes through the points of intersection of the parabolas \[{{y}^{2}}=4ax\] and \[{{x}^{2}}=4ay\], then [AIEEE 2004]
A)
\[{{d}^{2}}+{{(3b-2c)}^{2}}=0\] done
clear
B)
\[{{d}^{2}}+{{(3b+2c)}^{2}}=0\] done
clear
C)
\[{{d}^{2}}+{{(2b-3c)}^{2}}=0\] done
clear
D)
\[{{d}^{2}}+{{(2b+3c)}^{2}}=0\] done
clear
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question_answer15)
The locus of mid point of that chord of parabola which subtends right angle on the vertex will be [UPSEAT 1999]
A)
\[{{y}^{2}}-2ax+8{{a}^{2}}=0\] done
clear
B)
\[{{y}^{2}}=a(x-4a)\] done
clear
C)
\[{{y}^{2}}=4a(x-4a)\] done
clear
D)
\[{{y}^{2}}+3ax+4{{a}^{2}}=0\] done
clear
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question_answer16)
The equation of a circle passing through the vertex and the extremities of the latus rectum of the parabola \[{{y}^{2}}=8x\] is [MP PET 1998]
A)
\[{{x}^{2}}+{{y}^{2}}+10x=0\] done
clear
B)
\[{{x}^{2}}+{{y}^{2}}+10y=0\] done
clear
C)
\[{{x}^{2}}+{{y}^{2}}-10x=0\] done
clear
D)
\[{{x}^{2}}+{{y}^{2}}-5x=0\] done
clear
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question_answer17)
The centre of an ellipse is C and PN is any ordinate and A, A? are the end points of major axis, then the value of \[\frac{P{{N}^{2}}}{AN\ .\ A'N}\] is
A)
\[\frac{{{b}^{2}}}{{{a}^{2}}}\] done
clear
B)
\[\frac{{{a}^{2}}}{{{b}^{2}}}\] done
clear
C)
\[{{a}^{2}}+{{b}^{2}}\] done
clear
D)
1 done
clear
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question_answer18)
Let P be a variable point on the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] with foci \[{{F}_{1}}\] and \[{{F}_{2}}\]. If A is the area of the triangle \[P{{F}_{1}}{{F}_{2}}\], then maximum value of A is [IIT 1994; Kerala (Engg.) 2005]
A)
ab done
clear
B)
abe done
clear
C)
\[\frac{e}{ab}\] done
clear
D)
\[\frac{ab}{e}\] done
clear
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question_answer19)
A man running round a race-course notes that the sum of the distance of two flag-posts from him is always 10 metres and the distance between the flag-posts is 8 metres. The area of the path he encloses in square metres is [MNR 1991; UPSEAT 2000]
A)
\[15\pi \] done
clear
B)
\[12\pi \] done
clear
C)
\[18\pi \] done
clear
D)
\[8\pi \] done
clear
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question_answer20)
If the angle between the lines joining the end points of minor axis of an ellipse with its foci is \[{{x}^{2}}-{{y}^{2}}=25\], then the eccentricity of the ellipse is [IIT Screening 1997; Pb. CET 2001; DCE 2002]
A)
1/2 done
clear
B)
\[1/\sqrt{2}\] done
clear
C)
\[\sqrt{3}/2\] done
clear
D)
\[1/2\sqrt{2}\] done
clear
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question_answer21)
The eccentricity of an ellipse, with its centre at the origin, is \[\frac{1}{2}\]. If one of the directrices is \[x=4\], then the equation of the ellipse is [AIEEE 2004]
A)
\[4{{x}^{2}}+3{{y}^{2}}=1\] done
clear
B)
\[3{{x}^{2}}+4{{y}^{2}}=12\] done
clear
C)
\[4{{x}^{2}}+3{{y}^{2}}=12\] done
clear
D)
\[3{{x}^{2}}+4{{y}^{2}}=1\] done
clear
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question_answer22)
The line \[x\cos \alpha +y\sin \alpha =p\] will be a tangent to the conic \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], if [Roorkee 1978]
A)
\[{{p}^{2}}={{a}^{2}}{{\sin }^{2}}\alpha +{{b}^{2}}{{\cos }^{2}}\alpha \] done
clear
B)
\[{{p}^{2}}={{a}^{2}}+{{b}^{2}}\] done
clear
C)
\[{{p}^{2}}={{b}^{2}}{{\sin }^{2}}\alpha +{{a}^{2}}{{\cos }^{2}}\alpha \] done
clear
D)
None of these done
clear
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question_answer23)
The angle of intersection of ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] and circle \[{{x}^{2}}+{{y}^{2}}=ab\], is
A)
\[{{\tan }^{-1}}\left( \frac{a-b}{ab} \right)\] done
clear
B)
\[{{\tan }^{-1}}\left( \frac{a+b}{ab} \right)\] done
clear
C)
\[{{\tan }^{-1}}\left( \frac{a+b}{\sqrt{ab}} \right)\] done
clear
D)
\[{{\tan }^{-1}}\left( \frac{a-b}{\sqrt{ab}} \right)\] done
clear
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question_answer24)
On the ellipse \[4{{x}^{2}}+9{{y}^{2}}=1\], the points at which the tangents are parallel to the line \[8x=9y\] are [IIT 1999]
A)
\[\left( \frac{2}{5},\ \frac{1}{5} \right)\] done
clear
B)
\[\left( -\frac{2}{5},\ \frac{1}{5} \right)\] done
clear
C)
\[\left( -\frac{2}{5},\ -\frac{1}{5} \right)\] done
clear
D)
\[\left( \frac{2}{5},\ -\frac{1}{5} \right)\] done
clear
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question_answer25)
The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{5}=1\], is [IIT Screening 2003]
A)
27/4 sq. unit done
clear
B)
9 sq. unit done
clear
C)
27/2 sq. unit done
clear
D)
27 sq. unit done
clear
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question_answer26)
Tangent is drawn to ellipse \[\frac{{{x}^{2}}}{27}+{{y}^{2}}=1\] at \[(3\sqrt{3}\cos \theta ,\ \sin \theta )\] where \[\theta \in (0,\ \pi /2)\]. Then the value of \[\theta \] such that sum of intercepts on axes made by this tangent is minimum, is [IIT Screening 2003]
A)
\[\pi /3\] done
clear
B)
\[\pi /6\] done
clear
C)
\[\pi /8\] done
clear
D)
\[\pi /4\] done
clear
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question_answer27)
The locus of the middle point of the intercept of the tangents drawn from an external point to the ellipse \[{{x}^{2}}+2{{y}^{2}}=2\] between the co-ordinates axes, is [IIT Screening 2004]
A)
\[\frac{1}{{{x}^{2}}}+\frac{1}{2{{y}^{2}}}=1\] done
clear
B)
\[\frac{1}{4{{x}^{2}}}+\frac{1}{2{{y}^{2}}}=1\] done
clear
C)
\[\frac{1}{2{{x}^{2}}}+\frac{1}{4{{y}^{2}}}=1\] done
clear
D)
\[\frac{1}{2{{x}^{2}}}+\frac{1}{{{y}^{2}}}=1\] done
clear
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question_answer28)
If the normal at any point P on the ellipse cuts the major and minor axes in G and g respectively and C be the centre of the ellipse, then [Kurukshetra CEE 1998]
A)
\[{{a}^{2}}{{(CG)}^{2}}+{{b}^{2}}{{(Cg)}^{2}}={{({{a}^{2}}-{{b}^{2}})}^{2}}\] done
clear
B)
\[{{a}^{2}}{{(CG)}^{2}}-{{b}^{2}}{{(Cg)}^{2}}={{({{a}^{2}}-{{b}^{2}})}^{2}}\] done
clear
C)
\[{{a}^{2}}{{(CG)}^{2}}-{{b}^{2}}{{(Cg)}^{2}}={{({{a}^{2}}+{{b}^{2}})}^{2}}\] done
clear
D)
None of these done
clear
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question_answer29)
The locus of the poles of normal chords of an ellipse is given by [UPSEAT 2001]
A)
\[\frac{{{a}^{6}}}{{{x}^{2}}}+\frac{{{b}^{6}}}{{{y}^{2}}}={{({{a}^{2}}-{{b}^{2}})}^{2}}\] done
clear
B)
\[\frac{{{a}^{3}}}{{{x}^{2}}}+\frac{{{b}^{3}}}{{{y}^{2}}}={{({{a}^{2}}-{{b}^{2}})}^{2}}\] done
clear
C)
\[\frac{{{a}^{6}}}{{{x}^{2}}}+\frac{{{b}^{6}}}{{{y}^{2}}}={{({{a}^{2}}+{{b}^{2}})}^{2}}\] done
clear
D)
\[\frac{{{a}^{3}}}{{{x}^{2}}}+\frac{{{b}^{3}}}{{{y}^{2}}}={{({{a}^{2}}+{{b}^{2}})}^{2}}\] done
clear
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question_answer30)
If \[\theta \] and \[\varphi \] are eccentric angles of the ends of a pair of conjugate diameters of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], then \[\theta -\varphi \] is equal to
A)
\[\pm \frac{\pi }{2}\] done
clear
B)
\[\pm \pi \] done
clear
C)
0 done
clear
D)
None of thesew done
clear
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question_answer31)
If PQ is a double ordinate of hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] such that OPQ is an equilateral triangle, O being the centre of the hyperbola. Then the eccentricity e of the hyperbola satisfies [EAMCET 1999]
A)
\[1<e<2/\sqrt{3}\] done
clear
B)
\[e=2/\sqrt{3}\] done
clear
C)
\[e=\sqrt{3}/2\] done
clear
D)
\[e>2/\sqrt{3}\] done
clear
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question_answer32)
Equation \[\frac{1}{r}=\frac{1}{8}+\frac{3}{8}\cos \theta \] represents [EAMCET 2002]
A)
A rectangular hyperbola done
clear
B)
A hyperbola done
clear
C)
An ellipse done
clear
D)
A parabola done
clear
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question_answer33)
If the two tangents drawn on hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] in such a way that the product of their gradients is \[{{c}^{2}}\], then they intersects on the curve
A)
\[{{y}^{2}}+{{b}^{2}}={{c}^{2}}({{x}^{2}}-{{a}^{2}})\] done
clear
B)
\[{{y}^{2}}+{{b}^{2}}={{c}^{2}}({{x}^{2}}+{{a}^{2}})\] done
clear
C)
\[a{{x}^{2}}+b{{y}^{2}}={{c}^{2}}\] done
clear
D)
None of these done
clear
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question_answer34)
C the centre of the hyperbola\[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\]. The tangents at any point P on this hyperbola meets the straight lines \[bx-ay=0\]and \[bx+ay=0\] in the points Q and R respectively. Then \[CQ\ .\ CR=\]
A)
\[{{a}^{2}}+{{b}^{2}}\] done
clear
B)
\[{{a}^{2}}-{{b}^{2}}\] done
clear
C)
\[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}\] done
clear
D)
\[\frac{1}{{{a}^{2}}}-\frac{1}{{{b}^{2}}}\] done
clear
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question_answer35)
If \[x=9\] is the chord of contact of the hyperbola \[{{x}^{2}}-{{y}^{2}}=9\], then the equation of the corresponding pair of tangents is [IIT 1999]
A)
\[9{{x}^{2}}-8{{y}^{2}}+18x-9=0\] done
clear
B)
\[9{{x}^{2}}-8{{y}^{2}}-18x+9=0\] done
clear
C)
\[9{{x}^{2}}-8{{y}^{2}}-18x-9=0\] done
clear
D)
\[9{{x}^{2}}-8{{y}^{2}}+18x+9=0\] done
clear
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question_answer36)
Let \[P(a\sec \theta ,\ b\tan \theta )\] and \[Q(a\sec \varphi ,\ b\tan \varphi )\], where \[\theta +\varphi =\frac{\pi }{2}\], be two points on the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\]. If (h, k) is the point of intersection of the normals at P and Q, then k is equal to [IIT 1999; MP PET 2002]
A)
\[\frac{{{a}^{2}}+{{b}^{2}}}{a}\] done
clear
B)
\[-\left( \frac{{{a}^{2}}+{{b}^{2}}}{a} \right)\] done
clear
C)
\[\frac{{{a}^{2}}+{{b}^{2}}}{b}\] done
clear
D)
\[-\left( \frac{{{a}^{2}}+{{b}^{2}}}{b} \right)\] done
clear
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question_answer37)
The combined equation of the asymptotes of the hyperbola \[2{{x}^{2}}+5xy+2{{y}^{2}}+4x+5y=0\] [Karnataka CET 2002]
A)
\[2{{x}^{2}}+5xy+2{{y}^{2}}=0\] done
clear
B)
\[2{{x}^{2}}+5xy+2{{y}^{2}}-4x+5y+2=0\] done
clear
C)
\[2{{x}^{2}}+5xy+2{{y}^{2}}+4x+5y-2=0\] done
clear
D)
\[2{{x}^{2}}+5xy+2{{y}^{2}}+4x+5y+2=0\] done
clear
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question_answer38)
An ellipse has eccentricity \[\frac{1}{2}\] and one focus at the point\[P\left( \frac{1}{2},\ 1 \right)\]. Its one directrix is the common tangent nearer to the point P, to the circle \[{{x}^{2}}+{{y}^{2}}=1\] and the hyperbola\[{{x}^{2}}-{{y}^{2}}=1\]. The equation of the ellipse in the standard form, is [IIT 1996]
A)
\[\frac{{{(x-1/3)}^{2}}}{1/9}+\frac{{{(y-1)}^{2}}}{1/12}=1\] done
clear
B)
\[\frac{{{(x-1/3)}^{2}}}{1/9}+\frac{{{(y+1)}^{2}}}{1/12}=1\] done
clear
C)
\[\frac{{{(x-1/3)}^{2}}}{1/9}-\frac{{{(y-1)}^{2}}}{1/12}=1\] done
clear
D)
\[\frac{{{(x-1/3)}^{2}}}{1/9}-\frac{{{(y+1)}^{2}}}{1/12}=1\] done
clear
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question_answer39)
If a circle cuts a rectangular hyperbola \[xy={{c}^{2}}\] in A, B, C, D and the parameters of these four points be \[{{t}_{1}},\ {{t}_{2}},\ {{t}_{3}}\] and \[{{t}_{4}}\] respectively. Then [Kurukshetra CEE 1998]
A)
\[{{t}_{1}}{{t}_{2}}={{t}_{3}}{{t}_{4}}\] done
clear
B)
\[{{t}_{1}}{{t}_{2}}{{t}_{3}}{{t}_{4}}=1\] done
clear
C)
\[{{t}_{1}}={{t}_{2}}\] done
clear
D)
\[{{t}_{3}}={{t}_{4}}\] done
clear
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question_answer40)
The equation of common tangents to the parabola \[{{y}^{2}}=8x\] and hyperbola \[3{{x}^{2}}-{{y}^{2}}=3\], is
A)
\[2x\pm y+1=0\] done
clear
B)
\[2x\pm y-1=0\] done
clear
C)
\[x\pm 2y+1=0\] done
clear
D)
\[x\pm 2y-1=0\] done
clear
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