Question 1 :
If the length of minor axis of an ellipse is equal to the distance between the foci then its eccentricity is:
Question 4 :
Given two fixed points $\mathrm{A}$and $\mathrm{B}$ and $\mathrm{A}\mathrm{B}=6$, then the simplest form of the equation to the locus of $\mathrm{P}$ such that $\mathrm{P}\mathrm{A}+\mathrm{P}\mathrm{B}=8$ is:<br/>
Question 5 :
If the chords of the rectangular hyperbola ${ x }^{ 2 }-{ y }^{ 2 }={ a }^{ 2 }$ touch the parabola ${ y }^{ 2 }=4ax$, then the locus of their mid-points is <br>
Question 6 :
The two parabolas $ x^2 = 4y $ and $ y^2 = 4x $ meet in two distinct points. One of these is the origin and the other is :
Question 7 :
The point intersection of the two tangents at the ends of the latus rectum of the parabola $ (y+3)^2=8(x-2)$ is
Question 8 :
Assertion: Point $(5,5)$ lies inside the hyperbola $7x^2-2y^2+12xy-2x+14y-22=0$
Reason: Let $S=7x^2-2y^2+12xy-2x+14y-22$ and P$(5,5)$ then $S_1<0$
Question 9 :
Assertion: Statement - 1: If the foci of an hyperbola are at points $(4, 1)$ and $(-6, 1)$, eccentricity is $\dfrac54$, then the length of the transverse axis is $4$.
Reason: Statement - 2: Distance between the foci of a hyperbola is equal to the product of its eccentricity and the length of the transverse axis.
Question 10 :
The locus of a point P, whose distance from the point (1, -2) is always two times its distance from y-axis is.
Question 11 :
The number of values of $c$ such that the straight line $y = 4x + c$ touches the curve $\dfrac{x^2}{ 4}\,+\, y^2\,=\,1$ is
Question 12 :
The equation to the locus of the middle points of the portion of the tangent to the ellipse $\displaystyle {\frac{x^2}{16}\, +\, \frac{y^2}{9}\, =\, 1}$ included between the co-ordinate axes is the curve.
Question 13 :
The straight line $2x+y+c=0$ is a tangent to the ellipse $4{ x }^{ 2 }+8{ y }^{ 2 }=32$ if $c$ is
Question 14 :
$Assertion\ (A):$ The eccentricity of an ellipse is $\displaystyle \frac{3}{5}.$<br/>$Reason\ (R):$ The equation of the ellipse is $x=5\cos\theta, y=4\sin\theta.$ <br/>
Question 15 :
A hyperbola having the transverse axis of length $\sqrt{2}$ is confocal with $3x^2 + 4y^2 = 12$, then its equation is:<br/>
Question 16 :
If the normal to the given hyperbola at the point $(ct, \dfrac{c}{t})$ meets the curve again at $(ct', \dfrac{c}{t'})$, then
Question 17 :
Equation of the two tangents drawn from $(2,-1)$ to $\mathrm{x}^{2}+3\mathrm{y}^{2}=3$ are:<br/>
Question 18 :
Locus of middle point of all chords of $\displaystyle \frac { { x }^{ 2 } }{ 4 } +\frac { { y }^{ 2 } }{ 9 } =1$ which are at distance $2$ units from vertex of the parabola ${ y }^{ 2 }=-8ax$ is:
Question 20 :
The number of real tangents that can be drawn to the ellipse $3{ x }^{ 2 }+5{ y }^{ 2 }=32$ and $25{ x }^{ 2 }+9{ y }^{ 2 }=450$ passing through $(3,5)$ is
Question 21 :
Tangents are drawn from a point $P\left( {6,\,\sqrt 5 } \right)$ to the ellipse $\dfrac{{{x^2}}}{{25}} + \dfrac{{{y^2}}}{{16}} = 1$ touching the ellipse in the points $Q$ and $R$. The angle between $PQ$ and $PR$ is
Question 22 :
The Vertex of the parabola $y^{2} - 10y + x + 22=0$ is.
Question 24 :
Assertion: Ellipse $\displaystyle \frac{x^2}{25}\, +\, \frac{y^2}{16}\, =\, 1$ and $12x^2\, -\, 4y^2\, =\, 27$ intersect each other at right angle.
Reason: Whenever confocal ellipse & hyperbola intersect, they intersect each other orthogonally.
Question 26 :
The line $\displaystyle x-1= 0$ is the directrix of the parabola $\displaystyle y^{2}-kx+8= 0.$ Then one of the values of $k$ is
Question 27 :
AB is a double ordinate of the hyperbola $\displaystyle \frac{x^2}{a^2}\, -\, \frac{y^2}{b^2}\, =\, 1$ such that AOB (where 'O' is the origin) is an equilateral triangle, then the eccentricity $e$ of the hyperbola satisfies -
Question 28 :
An ellipse passing through the point $(2\sqrt{13},4)$ has its foci at $(-4,1)$ and $(4,1)$, then its eccentricity is <br>
Question 29 :
Equation of the transverse axis of the hyperbola $\displaystyle \dfrac{(y-2)^{2}}{9}-\dfrac{(x+3)^{2}}{16}=1$ is<br>
Question 30 :
The foci of an ellipse are $S(-1, -1), S'(0, -2)$ and its $\mathrm{e}=\dfrac12$, then the equation of the directrix corresponding to the focus $S$ is :<br>
Question 31 :
Assertion: The equation $\displaystyle 3x^{2} - 2y^{2} + 4x - 6y = 0$ represents a hyperbola.
Reason: The second degree equation $\displaystyle ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0$ represents a hyperbola if $\displaystyle abc + 2fgh - af^{2} - bg^{2} - ch^{2} \neq 0$ & $\displaystyle h^{2} > ab$.
Question 33 :
The perimeter of a triangle is $20$ and the points $(-2,-3)$ and $(-2,3)$ are two of the vertices of it. Then the locus of third vertex is:
Question 34 :
The locus of the foot of perpendicular from the focus on any tangent to the hyperbola $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ is:
Question 35 :
Assertion: Two mutually perpendicular tangents to the parabola $x^{2}=6y$ meet at point $\left ( -1,-\dfrac{3}{2} \right )$.
Reason: The tangents drawn from any point on directrix to the parabola, are perpendicular to each other.
Question 36 :
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
Question 38 :
The latus rectum of the hyperbola $\displaystyle \frac{x^{2}}{16}-\frac{y^{2}}{b}=1$ is $\displaystyle 4\frac{1}{2}$. Its eccentricity $\mathrm{e}=$<br/>
Question 40 :
On the parabola $y^2=64x,$ find the point nearest to the straight line $4x+3y-14=0$.
Question 41 :
Equation to the locus of the point which moves such that the sum of its distances from $(-4,3)$ and $(4,3)$ is $12$ is
Question 42 :
The equation of the common tangent to the parabola $y^2\, =\, 8x$ and the hyperbola $3x^2\, \,- y^2\, =\, 3$ is
Question 43 :
The curve described parametrically $x = t^{2} + t + 1, y = t^{2} - t + 1$ represents
Question 44 :
If $y = a ln |x| + bx^2 + x$ has its extreme values at $x = -1$ and $x = 2$ then P (a , b) is<br>
Question 45 :
$P$ is a point on the hyperbola $\dfrac { { x }^{ 2 } }{ { a }^{ 2 } } -\dfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1$, $N$ is the foot of the perpendicular from $P$ on the transverse axis. The tangent to the hyperbola at $P$ meets the transverse axis at $T$. If $O$ is the centre of the hyperbola, then $OT$. $ON$ is equal to:
Question 46 :
The equation of the hyperbola with eccentricity $\displaystyle\frac{3}{2}$ and foci at $(\pm2,0)$
Question 47 :
Tangents to the ellipse $\mathrm{b}^{2}\mathrm{x}^{2}+\mathrm{a}^{2}\mathrm{y}^{2}=\mathrm{a}^{2}\mathrm{b}^{2}$ make complementary angles with the major axis. Then the locus of their point of intersection is<br/>
Question 48 :
<b></b>The locus of the foot of the perpendicular from the centre of the hyperbola $xy\, =\, c^2$ on a variable tangent is :
Question 49 :
If the foci of ellipse $\displaystyle \frac{x^2}{25}\, +\, \displaystyle \frac{y^2}{16}\, =\, 1$ and the hyperbola $\displaystyle \frac{x^2}{4}\, -\, \displaystyle \frac{y^2}{b^2}\, =\, 1$ coincide, then $b^2$ =
