CONIC SECTIONS - Pyqs

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Q1
The eccentricity of the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) is
(Exam: IIT-JEE Year: 1998)
(A) \(\frac{\sqrt{5}}{3}\)
(B) \(\frac{\sqrt{5}}{2}\)
(C) \(\frac{2}{3}\)
(D) \(\frac{1}{\sqrt{5}}\)
Q2
The focus of the parabola \(y^2=8x\) is
(Exam: AIEEE Year: 2005)
(A) \((2,0)\)
(B) \((4,0)\)
(C) \((0,2)\)
(D) \((0,4)\)
Q3
The equation of the directrix of the parabola \(x^2=12y\) is
(Exam: NEET Year: 2016)
(A) \(y=-3\)
(B) \(y=3\)
(C) \(x=-3\)
(D) \(x=3\)
Q4
The length of the latus rectum of the ellipse \(\frac{x^2}{25}+\frac{y^2}{9}=1\) is
(Exam: IIT-JEE Year: 2002)
(A) \(\frac{18}{5}\)
(B) \(\frac{25}{3}\)
(C) \(6\)
(D) \(10\)
Q5
The asymptotes of the hyperbola \(x^2-y^2=9\) are
(Exam: IIT-JEE Year: 1996)
(A) \(y=\pm x\)
(B) \(y=\pm \frac{x}{3}\)
(C) \(y=\pm 3x\)
(D) \(y=\pm \sqrt{3}x\)
Q6
The centre of the circle \(x^2+y^2-4x+6y+9=0\) is
(Exam: BITSAT Year: 2012)
(A) \((2,-3)\)
(B) \((-2,3)\)
(C) \((4,-6)\)
(D) \((-4,6)\)
Q7
The eccentricity of the hyperbola \(\frac{x^2}{16}-\frac{y^2}{9}=1\) is
(Exam: IIT-JEE Year: 2000)
(A) \(\frac{5}{4}\)
(B) \(\frac{4}{5}\)
(C) \(\sqrt{\frac{7}{9}}\)
(D) \(\sqrt{\frac{25}{16}}\)
Q8
The vertex of the parabola \(y^2-4y-8x+12=0\) is
(Exam: KVPY Year: 2014)
(A) \((1,2)\)
(B) \((2,1)\)
(C) \((0,2)\)
(D) \((2,0)\)
Q9
The equation of the auxiliary circle of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) is
(Exam: IIT-JEE Year: 1999)
(A) \(x^2+y^2=a^2\)
(B) \(x^2+y^2=b^2\)
(C) \(x^2+y^2=a^2+b^2\)
(D) \(x^2+y^2=1\)
Q10
The focus of the hyperbola \(y^2-x^2=16\) is
(Exam: AIEEE Year: 2008)
(A) \((0,4\sqrt{2})\)
(B) \((4\sqrt{2},0)\)
(C) \((0,4)\)
(D) \((4,0)\)
Q11
The equation of the director circle of the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) is
(Exam: IIT-JEE Year: 2003)
(A) \(x^2+y^2=13\)
(B) \(x^2+y^2=9\)
(C) \(x^2+y^2=4\)
(D) \(x^2+y^2=5\)
Q12
The latus rectum of the parabola \(y^2=4x\) has length
(Exam: NEET Year: 2018)
(A) \(4\)
(B) \(2\)
(C) \(1\)
(D) \(8\)
Q13
The eccentricity of a circle is
(Exam: Olympiad Year: 2010)
(A) \(0\)
(B) \(1\)
(C) \(<1\)
(D) \(>1\)
Q14
The equation of the tangent to the circle \(x^2+y^2=25\) at point \((3,4)\) is
(Exam: IIT-JEE Year: 1995)
(A) \(3x+4y=25\)
(B) \(4x+3y=25\)
(C) \(3x+4y=5\)
(D) \(x+ y=5\)
Q15
The number of foci of a hyperbola is
(Exam: BITSAT Year: 2011)
(A) \(1\)
(B) \(2\)
(C) \(3\)
(D) Infinite
Q16
The standard equation of a parabola opening downward with vertex at origin is
(Exam: AIEEE Year: 2006)
(A) \(y^2=4ax\)
(B) \(x^2=4ay\)
(C) \(x^2=-4ay\)
(D) \(y^2=-4ax\)
Q17
The transverse axis length of the hyperbola \(\frac{x^2}{25}-\frac{y^2}{16}=1\) is
(Exam: IIT-JEE Year: 2001)
(A) \(10\)
(B) \(8\)
(C) \(5\)
(D) \(4\)
Q18
The equation represents a parabola if
(Exam: NEET Year: 2019)
(A) \(ax^2+by^2+dx+ey+f=0,\ a=b\)
(B) \(ax^2+by^2+dx+ey+f=0,\ ab<0\)
(C) \(ax^2+by^2+dx+ey+f=0,\ a=0\)
(D) \(ax^2+by^2+dx+ey+f=0,\ ab>0\)
Q19
The focus of the ellipse \(\frac{x^2}{16}+\frac{y^2}{12}=1\) is
(Exam: IIT-JEE Year: 2004)
(A) \((2,0)\)
(B) \((\sqrt{4},0)\)
(C) \((\sqrt{4},\sqrt{3})\)
(D) \((\sqrt{4},0)\)
Q20
The equation of the circle passing through origin with centre \((2,3)\) is
(Exam: AIEEE Year: 2009)
(A) \(x^2+y^2-4x-6y=0\)
(B) \(x^2+y^2+4x+6y=0\)
(C) \(x^2+y^2-2x-3y=0\)
(D) \(x^2+y^2+2x+3y=0\)
Q21
The eccentricity of a rectangular hyperbola is
(Exam: IIT-JEE Year: 1994)
(A) \(1\)
(B) \(\sqrt{2}\)
(C) \(2\)
(D) \(\frac{1}{\sqrt{2}}\)
Q22
The focal length of the parabola \(x^2=20y\) is
(Exam: NEET Year: 2020)
(A) \(5\)
(B) \(10\)
(C) \(20\)
(D) \(4\)
Q23
The minor axis length of the ellipse \(\frac{x^2}{36}+\frac{y^2}{16}=1\) is
(Exam: IIT-JEE Year: 2006)
(A) \(8\)
(B) \(6\)
(C) \(4\)
(D) \(12\)
Q24
The number of asymptotes of a hyperbola is
(Exam: BITSAT Year: 2010)
(A) \(1\)
(B) \(2\)
(C) \(3\)
(D) \(4\)
Q25
The equation of the directrix of the parabola \(y^2=-16x\) is
(Exam: IIT-JEE Year: 1997)
(A) \(x=4\)
(B) \(x=-4\)
(C) \(y=4\)
(D) \(y=-4\)
Q26
The distance between the foci of the ellipse \(\frac{x^2}{25}+\frac{y^2}{9}=1\) is
(Exam: NEET Year: 2017)
(A) \(8\)
(B) \(6\)
(C) \(10\)
(D) \(4\)
Q27
The equation of the hyperbola with asymptotes \(y=\pm 2x\) is
(Exam: IIT-JEE Year: 2008)
(A) \(x^2-4y^2=1\)
(B) \(4x^2-y^2=1\)
(C) \(y^2-4x^2=1\)
(D) \(y^2-2x^2=1\)
Q28
The standard equation of a circle of radius \(r\) and centre \((h,k)\) is
(Exam: Olympiad Year: 2009)
(A) \((x+h)^2+(y+k)^2=r^2\)
(B) \((x-h)^2+(y-k)^2=r^2\)
(C) \(x^2+y^2+hx+ky=r^2\)
(D) \(x^2+y^2=r^2\)
Q29
The slope of the tangent to \(y^2=4ax\) at point \((at^2,2at)\) is
(Exam: IIT-JEE Year: 2005)
(A) \(t\)
(B) \(\frac{1}{t}\)
(C) \(2t\)
(D) \(\frac{1}{2t}\)
Q30
The conic with eccentricity equal to one is
(Exam: NEET Year: 2021)
(A) Ellipse
(B) Circle
(C) Hyperbola
(D) Parabola
Q31
The equation of the normal to the parabola \(y^2=4ax\) at parameter \(t\) is
(Exam: IIT-JEE Year: 2007)
(A) \(y=-tx+2at+at^3\)
(B) \(y=tx-at^2\)
(C) \(y=-\frac{1}{t}x+2a+at^2\)
(D) \(y=tx+2at\)
Q32
The length of the focal chord through the vertex of the parabola \(y^2=12x\) is
(Exam: NEET Year: 2015)
(A) \(12\)
(B) \(6\)
(C) \(24\)
(D) \(3\)
Q33
The equation represents a pair of straight lines if
(Exam: IIT-JEE Year: 1993)
(A) \(\Delta=0\)
(B) \(\Delta>0\)
(C) \(\Delta<0\)
(D) \(\Delta\neq0\)
Q34
The parametric coordinates of any point on the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) are
(Exam: IIT-JEE Year: 1998)
(A) \((a\sec\theta,b\tan\theta)\)
(B) \((a\cos\theta,b\sin\theta)\)
(C) \((a\tan\theta,b\sec\theta)\)
(D) \((a\sin\theta,b\cos\theta)\)
Q35
The locus of a point whose distance from focus equals its distance from directrix is
(Exam: NEET Year: 2014)
(A) Ellipse
(B) Hyperbola
(C) Parabola
(D) Circle
Q36
The equation of the chord of contact of tangents drawn from \((x_1,y_1)\) to the circle \(x^2+y^2=a^2\) is
(Exam: IIT-JEE Year: 2004)
(A) \(xx_1+yy_1=a^2\)
(B) \(x^2+y^2=a^2\)
(C) \(x_1^2+y_1^2=a^2\)
(D) \(x^2+y^2=xx_1+yy_1\)
Q37
The condition for the line \(y=mx+c\) to be tangent to the parabola \(y^2=4ax\) is
(Exam: IIT-JEE Year: 2001)
(A) \(c=am^2\)
(B) \(c=-am^2\)
(C) \(c=2am\)
(D) \(c=a\)
Q38
The eccentricity of the ellipse whose directrix is at a distance \(10\) from centre and focus at distance \(6\) is
(Exam: AIIMS Year: 2010)
(A) \(\frac{3}{5}\)
(B) \(\frac{5}{3}\)
(C) \(\frac{2}{3}\)
(D) \(\frac{3}{2}\)
Q39
The pair of lines represented by \(x^2-y^2=0\) are
(Exam: Olympiad Year: 2008)
(A) \(x=y\) and \(x=-y\)
(B) \(x=0\) and \(y=0\)
(C) \(y=2x\) and \(y=-2x\)
(D) \(x+y=0\) and \(x-y=0\)
Q40
The equation of the normal to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) at \((a\cos\theta,b\sin\theta)\) is
(Exam: IIT-JEE Year: 2006)
(A) \(ax\sin\theta-by\cos\theta=(a^2-b^2)\sin\theta\cos\theta\)
(B) \(\frac{x\cos\theta}{a}+\frac{y\sin\theta}{b}=1\)
(C) \(bx\cos\theta+ay\sin\theta=ab\)
(D) \(ax\cos\theta+by\sin\theta=a^2\)
Q41
The director circle of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) is
(Exam: IIT-JEE Year: 1997)
(A) \(x^2+y^2=a^2-b^2\)
(B) \(x^2+y^2=a^2+b^2\)
(C) \(x^2-y^2=a^2\)
(D) \(x^2+y^2=ab\)
Q42
The slope of the tangent to the hyperbola \(\frac{x^2}{16}-\frac{y^2}{9}=1\) at parameter \(t\) is
(Exam: IIT-JEE Year: 2009)
(A) \(\frac{4t}{3}\)
(B) \(\frac{3t}{4}\)
(C) \(\frac{4}{3t}\)
(D) \(\frac{3}{4t}\)
Q43
The locus of the midpoint of focal chord of parabola \(y^2=4ax\) is
(Exam: IIT-JEE Year: 2010)
(A) \(x=a\)
(B) \(y=0\)
(C) \(x+y=a\)
(D) \(x=-a\)
Q44
The equation \(ax^2+2hxy+by^2=0\) represents a rectangular hyperbola if
(Exam: IIT-JEE Year: 1994)
(A) \(a=b\)
(B) \(h=0\)
(C) \(a+b=0\)
(D) \(ab=h^2\)
Q45
The latus rectum of the hyperbola \(\frac{y^2}{25}-\frac{x^2}{9}=1\) has length
(Exam: NEET Year: 2019)
(A) \(\frac{18}{5}\)
(B) \(\frac{50}{3}\)
(C) \(10\)
(D) \(6\)
Q46
The condition that the equation \(2x^2+ky^2+6x-4y+1=0\) represents a parabola is
(Exam: BITSAT Year: 2013)
(A) \(k=0\)
(B) \(k=2\)
(C) \(k=-2\)
(D) \(k\neq2\)
Q47
The chord of the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) whose midpoint is \((1,1)\) is
(Exam: IIT-JEE Year: 2005)
(A) \( \frac{x}{9}+\frac{y}{4}= \frac{1}{9}+\frac{1}{4}\)
(B) \(xx_1/a^2+yy_1/b^2=1\)
(C) \(\frac{x}{9}+\frac{y}{4}=\frac{13}{36}\)
(D) \(\frac{x}{9}+\frac{y}{4}=1\)
Q48
The equation of the parabola whose focus is \((0,3)\) and directrix is \(y=-3\) is
(Exam: NEET Year: 2020)
(A) \(x^2=12y\)
(B) \(y^2=12x\)
(C) \(x^2=6y\)
(D) \(y^2=6x\)
Q49
The number of tangents that can be drawn from a point inside a circle is
(Exam: Olympiad Year: 2007)
(A) \(0\)
(B) \(1\)
(C) \(2\)
(D) Infinite
Q50
The conic represented by \(x^2+4y^2-6x+8y+9=0\) is
(Exam: IIT-JEE Year: 2003)
(A) Circle
(B) Parabola
(C) Ellipse
(D) Hyperbola

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