JEE Main & Advanced Mathematics Pair of Straight Lines Question Bank

done Angle between the pair of straight lines, Condition for parallel and perpendicular lines

• question_answer1) If the lines represented by the equation \[a{{x}^{2}}-bxy-{{y}^{2}}=0\] make angles \[\alpha \] and \[\beta \] with the x-axis, then \[\tan (\alpha +\beta )\]=

  A) \[\frac{b}{1+a}\].

  B) \[\frac{-b}{1+a}\]

  C) \[\frac{a}{1+b}\]

  D) None of these

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• question_answer2) Angle between the lines represented by the equation \[{{x}^{2}}+2xy\sec \theta +{{y}^{2}}=0\] is

  A) \[\theta \]

  B) \[2\theta \]

  C) \[\frac{\theta }{2}\]

  D) None of these

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• question_answer3) If the acute angles between the pairs of lines \[3{{x}^{2}}-7xy+4{{y}^{2}}=0\] and \[6{{x}^{2}}-5xy+{{y}^{2}}=0\] be \[{{\theta }_{1}}\] and \[{{\theta }_{2}}\] respectively, then

  A) \[{{\theta }_{1}}={{\theta }_{2}}\]

  B) \[{{\theta }_{1}}=2{{\theta }_{2}}\]

  C) \[2{{\theta }_{1}}={{\theta }_{2}}\]

  D) None of these

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• question_answer4) . The angle between the lines represented by the equation \[a{{x}^{2}}+xy+b{{y}^{2}}=0\] will be \[{{45}^{o}}\], if

  A) \[a=1,b=6\]

  B) \[a=1,b=-6\]

  C) \[a=6,b=1\]

  D) None of these

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• question_answer5) The lines represented by the equation \[9{{x}^{2}}+24xy+16{{y}^{2}}+21x+28y+6=0\] are

  A) Parallel

  B) Coincident

  C) Perpendicular

  D) None of these

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• question_answer6) Acute angle between the lines represented by \[({{x}^{2}}+{{y}^{2}})\sqrt{3}=4xy\] is [MP PET 1992]

  A) \[\pi /6\]

  B) \[\pi /4\]

  C) \[\pi /3\]

  D) None of these

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• question_answer7) Angles made by the lines represented by the equation \[xy+y=0\]with \[y-\]axis are

  A) \[{{0}^{o}}\]and \[{{90}^{o}}\]

  B) \[{{0}^{o}}\]and \[{{30}^{o}}\]

  C) \[{{30}^{o}}\]and \[{{60}^{o}}\]

  D) \[{{30}^{o}}\]and \[{{90}^{o}}\]

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• question_answer8) If the angle between the lines represented by the equation \[{{y}^{2}}+kxy-{{x}^{2}}\]\[{{\tan }^{2}}A=0\]be \[2A\], then \[k=\]

  A) 0

  B) 1

  C) 2

  D) \[\tan A\]

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• question_answer9) . The angle between the lines \[xy=0\] is [MP PET 1990, 92]

  A) \[{{45}^{o}}\]

  B) \[{{60}^{o}}\]

  C) \[{{90}^{o}}\]

  D) \[{{180}^{o}}\]

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• question_answer10) The angle between the lines represented by the equation \[4{{x}^{2}}-24xy+11{{y}^{2}}=0\] are [MP PET 1990]

  A) \[{{\tan }^{-1}}\frac{3}{4},{{\tan }^{-1}}\left( -\frac{3}{4} \right)\]

  B) \[{{\tan }^{-1}}\frac{1}{3},{{\tan }^{-1}}\left( -\frac{1}{3} \right)\]

  C) \[{{\tan }^{-1}}\frac{4}{3},{{\tan }^{-1}}\left( -\frac{4}{3} \right)\]

  D) \[{{\tan }^{-1}}\frac{1}{2},{{\tan }^{-1}}\left( -\frac{1}{2} \right)\]

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• question_answer11) If the lines represented by the equation \[2{{x}^{2}}-3xy+{{y}^{2}}=0\] make angles \[\alpha \]and \[\beta \] with x-axis, then \[{{\cot }^{2}}\alpha +{{\cot }^{2}}\beta \]=

  A) 0

  B) 3/2

  C) 7/4

  D) 5/4

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• question_answer12) Angle between the line joining the origin to the points of intersection of the curves \[2{{x}^{2}}+3{{y}^{2}}+10x=0\] and \[3{{x}^{2}}+5{{y}^{2}}+16x=0\] is

  A) \[{{\tan }^{-1}}\frac{3}{2}\]

  B) \[{{\tan }^{-1}}\frac{4}{5}\]

  C) \[{{90}^{o}}\]

  D) None of these

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• question_answer13) The lines \[{{(lx+my)}^{2}}-3{{(mx-ly)}^{2}}=0\] and \[lx+my+n=0\] form

  A) An isosceles triangle

  B) A right angled triangle

  C) An equilateral triangle

  D) None of these

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• question_answer14) The angle between the lines represented by the equation \[\lambda {{x}^{2}}+{{(1-\lambda )}^{2}}xy-\lambda {{y}^{2}}=0\], is

  A) \[{{30}^{o}}\]

  B) \[{{45}^{o}}\]

  C) \[{{60}^{o}}\]

  D) \[{{90}^{o}}\]

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• question_answer15) If the sum of the slopes of the lines represented by the equation \[{{x}^{2}}-2xy\tan A-{{y}^{2}}=0\]be 4, then \[\angle A=\]

  A) \[{{0}^{o}}\]

  B) \[{{45}^{o}}\]

  C) \[{{60}^{o}}\]

  D) \[{{\tan }^{-1}}(-2)\]

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• question_answer16) . If \[(a+3b)(3a+b)=4{{h}^{2}}\], then the angle between the lines represented by \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] is

  A) \[{{30}^{o}}\]

  B) \[{{45}^{o}}\]

  C) \[{{60}^{o}}\]

  D) \[{{\tan }^{-1}}\frac{1}{2}\]

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• question_answer17) The angle between the lines represented by the equation \[({{x}^{2}}+{{y}^{2}})\sin \theta +2xy=0\]is

  A) \[\theta \]

  B) \[\frac{\theta }{2}\]

  C) \[\frac{\pi }{2}-\theta \]

  D) \[\frac{\pi }{2}-\frac{\theta }{2}\]

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• question_answer18) The straight lines joining the origin to the points of intersection of the line \[2x+y=1\] and curve \[3{{x}^{2}}+4xy-4x+1=0\] include an angle [MP PET 1993]

  A) p/2

  B) p/3

  C) p/4

  D) p/6

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• question_answer19) The angle between the pair of straight lines \[{{x}^{2}}+4{{y}^{2}}-7xy=0\], is [MNR 1983]

  A) \[{{\tan }^{-1}}\left( \frac{1}{3} \right)\]

  B) \[{{\tan }^{-1}}3\]

  C) \[{{\tan }^{-1}}\frac{\sqrt{33}}{5}\]

  D) \[{{\tan }^{-1}}\frac{5}{\sqrt{33}}\]

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• question_answer20) The angle between the pair of lines given by equation \[{{x}^{2}}+2xy-{{y}^{2}}=0\], is [MNR 1990]

  A) p/3

  B) p/6

  C) p/2

  D) 0

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• question_answer21) The angle between the pair of straight lines \[{{x}^{2}}-{{y}^{2}}-2y-1=0\], is [MNR 1991; UPSEAT 2000]

  A) \[{{90}^{o}}\]

  B) \[{{60}^{o}}\]

  C) \[{{75}^{o}}\]

  D) \[{{36}^{o}}\]

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• question_answer22) If the angle between the two lines represented by \[2{{x}^{2}}+5xy+3{{y}^{2}}+6x+7y+4=0\] is \[{{\tan }^{-1}}m\], then \[m=\] [MNR 1993]

  A) 1/5

  B) 1

  C) 7/5

  D) 7

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• question_answer23) Pair of straight lines perpendicular to each other represented by [Roorkee 1990]

  A) \[2{{x}^{2}}=2y(2x+y)\]

  B) \[{{x}^{2}}+{{y}^{2}}+3=0\]

  C) \[2{{x}^{2}}=y(2x+y)\]

  D) \[{{x}^{2}}=2(x-y)\]

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• question_answer24) If the lines \[(p-q){{x}^{2}}+2(p+q)xy+(q-p){{y}^{2}}=0\] are mutually perpendicular, then

  A) \[p=q\]

  B) \[q=0\]

  C) \[p=0\]

  D) p and q may have any value

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• question_answer25) Which of the equation represents the pair of perpendicular straight lines

  A) \[{{y}^{2}}+xy-{{x}^{2}}=0\]

  B) \[{{y}^{2}}-xy+{{x}^{2}}=0\]

  C) \[{{x}^{2}}+xy+{{y}^{2}}=0\]

  D) \[{{x}^{2}}+xy-2{{y}^{2}}=0\]

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• question_answer26) The lines represented by the equation \[{{x}^{2}}+2\sqrt{3}xy+3{{y}^{2}}-3x-3\sqrt{3}y-4=0\], are

  A) Perpendicular to each other

  B) Parallel

  C) Inclined at \[{{45}^{o}}\]to each other

  D) None of these

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• question_answer27) If two of the three lines represented by the equation \[a{{x}^{3}}+b{{x}^{2}}y+cx{{y}^{2}}+d{{y}^{3}}=0\] are perpendicular, then

  A) \[{{a}^{2}}+{{d}^{2}}=2ac\]

  B) \[{{a}^{2}}+{{d}^{2}}=2bd\]

  C) \[{{a}^{2}}+ac+bd+{{d}^{2}}=0\]

  D) \[{{a}^{2}}+{{d}^{2}}=4bc\]

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• question_answer28) If in general quadratic equation \[f(x,y)=0,\Delta =0\] and \[a+b=0\], then the equation represents

  A) Two parallel straight lines

  B) Two perpendicular straight lines

  C) Two lines passing through origin

  D) None of these

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• question_answer29) The angle between the lines represented by the equation \[{{x}^{2}}-2pxy+{{y}^{2}}=0\], is

  A) \[{{\sec }^{-1}}p\]

  B) \[{{\cos }^{-1}}p\]

  C) \[{{\tan }^{-1}}p\]

  D) None of these

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• question_answer30) The angle between the lines represented by the equation \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\]is given by [RPET 1995]

  A) \[\tan \theta =\frac{2({{h}^{2}}-ab)}{(a+b)}\]

  B) \[\tan \theta =\frac{2\sqrt{{{h}^{2}}-ab}}{(a+b)}\]

  C) \[\tan \theta =\frac{2({{h}^{2}}-ab)}{\sqrt{a+b}}\]

  D) \[\tan \theta =\frac{2\sqrt{{{h}^{2}}+ab}}{(a+b)}\]

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• question_answer31) The equation \[{{x}^{2}}+{{k}_{1}}{{y}^{2}}+{{k}_{2}}xy=0\] represents a pair of perpendicular lines, if

  A) \[{{k}_{1}}=-1\]

  B) \[{{k}_{1}}=2{{k}_{2}}\]

  C) \[2{{k}_{1}}={{k}_{2}}\]

  D) None of these

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• question_answer32) The angle between the pair of lines \[2{{x}^{2}}+5xy+2{{y}^{2}}+3x+3y+1=0\] is [EAMCET 1994]

  A) \[{{\cos }^{-1}}\left( \frac{4}{5} \right)\]

  B) \[{{\tan }^{-1}}\left( \frac{4}{5} \right)\]

  C) 0

  D) p/2

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• question_answer33) The straight lines represented by the equation \[9{{x}^{2}}-12xy+4{{y}^{2}}=0\] are

  A) Coincident

  B) Perpendicular

  C) Parallel

  D) Inclined at an angle of\[{{45}^{o}}\]

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• question_answer34) The angle between the two straight lines \[2{{x}^{2}}-5xy+2{{y}^{2}}-3x+3y+1=0\] is

  A) \[{{45}^{o}}\]

  B) \[{{60}^{o}}\]

  C) \[{{\tan }^{-1}}\frac{4}{3}\]

  D) \[{{\tan }^{-1}}\frac{3}{4}\]

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• question_answer35) The acute angle formed between the lines joining the origin to the points of intersection of the curves \[{{x}^{2}}+{{y}^{2}}-2x-1=0\] and \[x+y=1\], is [MP PET 1998]

  A) \[{{\tan }^{-1}}\left( -\frac{1}{2} \right)\]

  B) \[{{\tan }^{-1}}2\]

  C) \[{{\tan }^{-1}}\frac{1}{2}\]

  D) \[{{60}^{o}}\]

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• question_answer36) The angle between the lines given by \[{{x}^{2}}-{{y}^{2}}=0\] is [MP PET 1999]

  A) \[{{15}^{o}}\]

  B) \[{{45}^{o}}\]

  C) \[{{75}^{o}}\]

  D) \[9{{0}^{o}}\]

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• question_answer37) Condition that the two lines represented by the equation \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\]to be perpendicular is [Kurukshetra CEE 1998; MP PET 2001]

  A) \[ab=-1\]

  B) \[a=-b\]

  C) \[a=b\]

  D) \[ab=1\]

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• question_answer38) The equation \[12{{x}^{2}}+7xy+a{{y}^{2}}+13x-y+3=0\] represents a pair of perpendicular lines. Then the value of ?a? is [Karnataka CET 2001]

  A) \[\frac{7}{2}\]

  B) - 19

  C) - 12

  D) 12

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• question_answer39) The angle between the lines \[{{x}^{2}}+4xy+{{y}^{2}}=0\] is [Karnataka CET 2001; Pb. CET 2001]

  A) \[{{60}^{o}}\]

  B) \[{{15}^{o}}\]

  C) \[{{30}^{o}}\]

  D) \[{{45}^{o}}\]

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• question_answer40) If the equation \[12{{x}^{2}}+7xy-p{{y}^{2}}-18x+qy+6=0\] represents a pair of perpendicular straight lines, then [Kurukshetra CEE 2002]

  A) \[p=12,q=1\]

  B) \[p=1,q=12\]

  C) \[p=-1,q=12\]

  D) \[p=1,q=-12\]

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• question_answer41) The angle between the pair of lines represented by \[2{{x}^{2}}-7xy+3{{y}^{2}}=0\], is [Kurukshetra CEE 2002]

  A) \[{{60}^{o}}\]

  B) \[{{45}^{o}}\]

  C) \[{{\tan }^{-1}}\left( \frac{7}{6} \right)\]

  D) \[{{30}^{o}}\]

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• question_answer42) If the angle \[2\theta \]is acute, then the acute angle between \[{{x}^{2}}(\cos \theta -\sin \theta )+2xy\cos \theta +{{y}^{2}}(\cos \theta +\sin \theta )=0\] is [EAMCET 2002]

  A) \[2\theta \]

  B) \[\theta /3\]

  C) \[\theta \]

  D) \[\theta /2\]

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• question_answer43) If the angle between the pair of straight lines represented by the equation \[{{x}^{2}}-3xy+\lambda {{y}^{2}}+3x-5y+2=0\] is \[{{\tan }^{-1}}\left( \frac{1}{3} \right)\], where \['\lambda \,'\]is a non negative real number. Then\[\lambda \]is [Orissa JEE 2002]

  A) 2

  B) 0

  C) 3

  D) 1

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• question_answer44) The angle between the lines \[{{x}^{2}}-xy-6{{y}^{2}}-7x+31y-18=0\] is [Karnataka CET 2003]

  A) \[{{45}^{o}}\]

  B) \[{{60}^{o}}\]

  C) \[{{90}^{o}}\]

  D) \[{{30}^{o}}\]

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