Total Questions:50 Total Time: 75 Min
Remaining:
Question:PQ is a double ordinate of the parabola \({y^2} = 4ax\). The locus of the points of trisection of PQ is
\(9{y^2} = 4ax\)
\(9{x^2} = 4ay\)
\(9{y^2} + 4ax = 0\)
\(9{x^2} + 4ay = 0\)
Question:If the vertex of a parabola be at origin and directrix be \(x + 5 = 0\), then its latus rectum is
5
10
20
40
Question:The equation of the parabola with focus (3, 0) and the directirx \(x + 3 = 0\) is
\({y^2} = 3x\)
\({y^2} = 2x\)
\({y^2} = 12x\)
\({y^2} = 6x\)
Question:Locus of the poles of focal chords of a parabola is of parabola
The tangent at the vertex
The axis
A focal chord
The directrix
Question:The equation of the parabola whose vertex and focus lies on the x-axis at distance a and a' from the origin, is
\({y^2} = 4(a' - a)(x - a)\)
\({y^2} = 4(a' - a)(x + a)\)
\({y^2} = 4(a' + a)(x - a)\)
\({y^2} = 4(a' + a)(x + a)\)
Question:The focus of the parabola \({y^2} = 4y - 4x\) is
(0, 2)
(1, 2)
(2, 0)
(2, 1)
Question:Vertex of the parabola \({x^2} + 4x + 2y - 7 = 0\) is
(–2, 11/2)
(–2, 2)
(–2, 11)
(2, 11)
Question:The focus of the parabola \({x^2} = 2x + 2y\) is
\(\left( {\frac{3}{2},\;\frac{{ - 1}}{2}} \right)\)
\(\left( {1,\;\frac{{ - 1}}{2}} \right)\)
(1, 0)
(0, 1)
Question:Latus rectum of the parabola \({y^2} - 4y - 2x - 8 = 0\) is
2
4
8
1
Question:The equation of the parabola with focus (a, b) and directrix \(\frac{x}{a} + \frac{y}{b} = 1\) is given by
\({(ax - by)^2} - 2{a^3}x - 2{b^3}y + {a^4} + {a^2}{b^2} + {b^4} = 0\)
\({(ax + by)^2} - 2{a^3}x - 2{b^3}y - {a^4} + {a^2}{b^2} - {b^4} = 0\)
\({(ax - by)^2} + {a^4} + {b^4} - 2{a^3}x = 0\)
\({(ax - by)^2} - 2{a^3}x = 0\)
Question:The equation of the parabola whose vertex is at (2, -1) and focus at (2, -3) is
\({x^2} + 4x - 8y - 12 = 0\)
\({x^2} - 4x + 8y + 12 = 0\)
\({x^2} + 8y = 12\)
\({x^2} - 4x + 12 = 0\)
Question:The directrix of the parabola \({x^2} - 4x - 8y + 12 = 0\) is
\(x = 1\)
\(y = 0\)
\(x = - 1\)
\(y = - 1\)
Question:The straight line \(y = 2x + \lambda \) does not meet the parabola \({y^2} = 2x\), if
\(\lambda < \frac{1}{4}\)
\(\lambda > \frac{1}{4}\)
\(\lambda = 4\)
\(\lambda = 1\)
Question:The equation of the tangent at a point \(P(t)\) where 't' is any parameter to the parabola \({y^2} = 4ax\), is
\(yt = x + a{t^2}\)
\(y = xt + a{t^2}\)
\(y = xt + \frac{a}{t}\)
\(y = tx\)
Question:The line \(y = 2x + c\) is tangent to the parabola \({y^2} = 4x\), then \(c = \)
\( - \frac{1}{2}\)
\(\frac{1}{2}\)
\(\frac{1}{3}\)
Question:The condition for which the straight line \(y = mx + c\) touches the parabola \({y^2} = 4ax\) is
\(a = c\)
\(\frac{a}{c} = m\)
\(m = {a^2}c\)
\(m = a{c^2}\)
Question:The angle of intersection between the curves \({x^2} = 4(y + 1)\) and \({x^2} = - 4(y + 1)\) is
\(\frac{\pi }{6}\)
\(\frac{\pi }{4}\)
0
\(\frac{\pi }{2}\)
Question:Angle between two curves \({y^2} = 4(x + 1)\) and \({x^2} = 4(y + 1)\) is
0 \(^{\rm{o}}\)
90 \(^{\rm{o}}\)
60 \(^{\rm{o}}\)
30 \(^{\rm{o}}\)
Question:The length of chord of contact of the tangents drawn from the point (2, 5) to the parabola \({y^2} = 8x\), is
\(\frac{1}{2}\sqrt {41} \)
\(\sqrt {41} \)
\(\frac{3}{2}\sqrt {41} \)
\(2\sqrt {41} \)
Question:If 'a' and 'c' are the segments of a focal chord of a parabola and b the semi-latus rectum, then
a, b, c are in A.P.
a, b, c are in G.P.
a, b, c are in H.P.
None of these
Question:The focal chord to \({y^2} = 16x\) is tangent to \({(x - 6)^2} + {y^2} = 2\), then the possible value of the slope of this chord, are
\(\{ - 1,\;1\} \)
{–2, 2}
{-2, 1/2}
{2, –1/2}
Question:The normal to the parabola \({y^2} = 8x\) at the point (2, 4) meets the parabola again at the point
{–18, –12}
{–18, 12}
{18, 12}
(18, –12)
Question:The equation to a parabola which passes through the intersection of a straight line \(x + y = 0\) and the circle \({x^2} + {y^2} + 4y = 0\) is
\({y^2} = 4x\)
\({y^2} = x\)
Question:Let a circle tangent to the directrix of a parabola \({y^2} = 2ax\) has its centre coinciding with the focus of the parabola. Then the point of intersection of the parabola and circle is
(a, –a)
\((a/2,\;a/2)\)
\((a/2,\; \pm a)\)
\(( \pm a,\;a/2)\)
Question:The equation of the ellipse whose vertices are \(( \pm 5,\;0)\) and foci are \(( \pm 4,\;0)\) is
\(9{x^2} + 25{y^2} = 225\)
\(25{x^2} + 9{y^2} = 225\)
\(3{x^2} + 4{y^2} = 192\)
Question:The equation of the ellipse whose foci are \(( \pm 5,\;0)\) and one of its directrix is \(5x = 36\), is
\(\frac{{{x^2}}}{{36}} + \frac{{{y^2}}}{{11}} = 1\)
\(\frac{{{x^2}}}{6} + \frac{{{y^2}}}{{\sqrt {11} }} = 1\)
\(\frac{{{x^2}}}{6} + \frac{{{y^2}}}{{11}} = 1\)
Question:If the length of the major axis of an ellipse is three times the length of its minor axis, then its eccentricity is
\(\frac{1}{{\sqrt 3 }}\)
\(\frac{1}{{\sqrt 2 }}\)
\(\frac{{2\sqrt 2 }}{3}\)
Question:The length of the latus rectum of an ellipse is \(\frac{1}{3}\) of the major axis. Its eccentricity is
\(\frac{2}{3}\)
\(\sqrt {\frac{2}{3}} \)
\(\frac{{5 \times 4 \times 3}}{{{7^3}}}\)
\({\left( {\frac{3}{4}} \right)^4}\)
Question:The eccentricity of the ellipse \(4{x^2} + 9{y^2} = 36\), is
\(\frac{1}{{2\sqrt 3 }}\)
\(\frac{{\sqrt 5 }}{3}\)
\(\frac{{\sqrt 5 }}{6}\)
Question:The eccentricity of the ellipse \(25{x^2} + 16{y^2} = 400\) is
3/5
1/3
2/5
1/5
Question:The equation \(14{x^2} - 4xy + 11{y^2} - 44x - 58y + 71 = 0\) represents
A circle
An ellipse
A hyperbola
A rectangular hyperbola
Question:The centre of the ellipse\(\frac{{{{(x + y - 2)}^2}}}{9} + \frac{{{{(x - y)}^2}}}{{16}} = 1\) is
(0, 0)
(1, 1)
Question:The position of the point (4, -3) with respect to the ellipse \(2{x^2} + 5{y^2} = 20\) is
Outside the ellipse
On the ellipse
On the major axis
Question:The equation of the tangent to the ellipse \({x^2} + 16{y^2} = 16\) making an angle of \({60^o}\)with x-axis is
\(\sqrt 3 x - y + 7 = 0\)
\(\sqrt 3 x - y - 7 = 0\)
\(\sqrt 3 x - y \pm 7 = 0\)
Question:The equation of the normal to the ellipse \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\) at the point \((a\cos \theta ,\;b\sin \theta )\) is
\(\frac{{ax}}{{\sin \theta }} - \frac{{by}}{{\cos \theta }} = {a^2} - {b^2}\)
\(\frac{{ax}}{{\sin \theta }} - \frac{{by}}{{\cos \theta }} = {a^2} + {b^2}\)
\(\frac{{ax}}{{\cos \theta }} - \frac{{by}}{{\sin \theta }} = {a^2} - {b^2}\)
\(\frac{{ax}}{{\cos \theta }} - \frac{{by}}{{\sin \theta }} = {a^2} + {b^2}\)
Question:If the normal at the point \(P(\theta )\) to the ellipse \(\frac{{{x^2}}}{{14}} + \frac{{{y^2}}}{5} = 1\) intersects it again at the point \(Q(2\theta )\), then \(\cos \theta \) is equal to
\( - \frac{2}{3}\)
\(\frac{3}{2}\)
\( - \frac{3}{2}\)
Question:The point (4, -3) with respect to the ellipse \(4{x^2} + 5{y^2} = 1\)
Lies on the curve
Is inside the curve
Is outside the curve
Is focus of the curve
Question:A point ratio of whose distance from a fixed point and line \(x = 9/2\) is always 2 : 3. Then locus of the point will be
Hyperbola
Ellipse
Parabola
Circle
Question:The locus of the point of intersection of the lines \(ax\sec \theta + by\tan \theta = a\) and \(ax\tan \theta + by\sec \theta = b\), where \(\theta \) is the parameter, is
A straight line
Question:If the centre, vertex and focus of a hyperbola be (0, 0), (4, 0) and (6, 0) respectively, then the equation of the hyperbola is
\(4{x^2} - 5{y^2} = 8\)
\(4{x^2} - 5{y^2} = 80\)
\(5{x^2} - 4{y^2} = 80\)
\(5{x^2} - 4{y^2} = 8\)
Question:The eccentricity of the hyperbola \(2{x^2} - {y^2} = 6\) is
\(\sqrt 2 \)
3
\(\sqrt 3 \)
Question:The distance between the foci of a hyperbola is double the distance between its vertices and the length of its conjugate axis is 6. The equation of the hyperbola referred to its axes as axes of co-ordinates is
\(3{x^2} - {y^2} = 3\)
\({x^2} - 3{y^2} = 3\)
\(3{x^2} - {y^2} = 9\)
\({x^2} - 3{y^2} = 9\)
Question:The eccentricity of the hyperbola \(5{x^2} - 4{y^2} + 20x + 8y = 4\) is
Question:The latus rectum of the hyperbola \(9{x^2} - 16{y^2} + 72x - 32y - 16 = 0\) is
\(\frac{9}{2}\)
\( - \frac{9}{2}\)
\(\frac{{32}}{3}\)
\( - \frac{{32}}{3}\)
Question:If the straight line \(x\cos \alpha + y\sin \alpha = p\) be a tangent to the hyperbola \(\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1\), then
\({a^2}{\cos ^2}\alpha + {b^2}{\sin ^2}\alpha = {p^2}\)
\({a^2}{\cos ^2}\alpha - {b^2}{\sin ^2}\alpha = {p^2}\)
\({a^2}{\sin ^2}\alpha + {b^2}{\cos ^2}\alpha = {p^2}\)
\({a^2}{\sin ^2}\alpha - {b^2}{\cos ^2}\alpha = {p^2}\)
Question:If the tangent on the point \((2\sec \varphi ,\;3\tan \varphi )\) of the hyperbola \(\frac{{{x^2}}}{4} - \frac{{{y^2}}}{9} = 1\) is parallel to \(3x - y + 4 = 0\), then the value of \(\phi \) is
\({45^o}\)
\({60^o}\)
\({30^o}\)
\({75^o}\)
Question:The equation of the normal to the hyperbola \(\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = 1\) at \(( - 4,\;0)\) is
\(y = x\)
\(x = 0\)
\(x = - y\)
Question:The eccentricity of the conjugate hyperbola of the hyperbola \({x^2} - 3{y^2} = 1\), is
\(\frac{2}{{\sqrt 3 }}\)
\(\frac{4}{3}\)
Question:If \(5{x^2} + \lambda {y^2} = 20\) represents a rectangular hyperbola, then \(\lambda \) equals
-5
Question:The equation of the hyperbola referred to the axis as axes of co-ordinate and whose distance between the foci is 16 and eccentricity is \(\sqrt 2 \), is
\({x^2} - {y^2} = 16\)
\({x^2} - {y^2} = 32\)
\({x^2} - 2{y^2} = 16\)
\({y^2} - {x^2} = 16\)