Ellipse

If the radius of the largest circle with centre (2,0) inscribed in the ellipse \(x^2+4y^2=36\) is r, then 12r\(^2\) is equal to :

Let the ellipse \(E:{x^2} + 9{y^2} = 9\) intersect the positive x and y-axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. I...

Let a circle of radius 4 be concentric to the ellipse \(15 x^{2}+19 y^{2}=285\). Then the common tangents are inclined to the minor axis of the ellipse at the angle :

Let the eccentricity of the ellipse \({x^2} + {a^2}{y^2} = 25{a^2}\) be b times the eccentricity of the hyperbola \({x^2} - {a^2}{y^2} = 5\), where a is the minimum distance between the curves y = ex and y = logex. Then $${a^2} + {1 \over {...

Let a line L pass through the point of intersection of the lines \(b x+10 y-8=0\) and \(2 x-3 y=0, \mathrm{~b} \in \mathbf{R}-\left\{\frac{4}{3}\right\}\). If the line \(\mathrm{L}\) also passes through the point \((1,1)\) and touches the c...

Let the tangents at the points \(\mathrm{P}\) and \(\mathrm{Q}\) on the ellipse \(\frac{x^{2}}{2}+\frac{y^{2}}{4}=1\) meet at the point \(R(\sqrt{2}, 2 \sqrt{2}-2)\). If \(\mathrm{S}\) is the focus of the ellipse on its negative major axis,...

Let the eccentricity of an ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\), \(a > b\), be \({1 \over 4}\). If this ellipse passes through the point \(\left( { - 4\sqrt {{2 \over 5}} ,3} \right)\), then \({a^2} + {b^2}\) is...

If the length of the latus rectum of the ellipse \(x^{2}+4 y^{2}+2 x+8 y-\lambda=0\) is 4 , and \(l\) is the length of its major axis, then \(\lambda+l\) is equal to ____________.

The locus of the mid point of the line segment joining the point (4, 3) and the points on the ellipse \({x^2} + 2{y^2} = 4\) is an ellipse with eccentricity :

If m is the slope of a common tangent to the curves \({{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1\) and \({x^2} + {y^2} = 12\), then \(12{m^2}\) is equal to :

The acute angle between the pair of tangents drawn to the ellipse \(2 x^{2}+3 y^{2}=5\) from the point \((1,3)\) is :

The line y = x + 1 meets the ellipse \({{{x^2}} \over 4} + {{{y^2}} \over 2} = 1\) at two points P and Q. If r is the radius of the circle with PQ as diameter then (3r)2 is equal to :

If the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) meets the line \(\frac{x}{7}+\frac{y}{2 \sqrt{6}}=1\) on the \(x\)-axis and the line \(\frac{x}{7}-\frac{y}{2 \sqrt{6}}=1\) on the \(y\)-axis, then the eccentricity of the ellipse...

If two tangents drawn from a point (\(\alpha\), \(\beta\)) lying on the ellipse 25x2 + 4y2 = 1 to the parabola y2 = 4x are such that the slope of one tangent is four times the other, then the value of (10\(\alpha\) + 5)2 + (16\(\beta\)2 + 5...

Let the maximum area of the triangle that can be inscribed in the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over 4} = 1,\,a > 2\), having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to t...

The line \(12x\cos \theta + 5y\sin \theta = 60\) is tangent to which of the following curves?

An angle of intersection of the curves, \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\) and x2 + y2 = ab, a > b, is :

The locus of mid-points of the line segments joining (\(-\)3, \(-\)5) and the points on the ellipse \({{{x^2}} \over 4} + {{{y^2}} \over 9} = 1\) is :

A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix of an ellipse with eccentricity \({1 \over 3}\) and the distance of the nearer focus from...

Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its center at (3, \(-\)4), one focus at (4, \(-\)4) and one vertex at (5, \(-\)4). If mx \(-\) y = 4, m > 0 is a tangent to the ellipse E, then the value of 5m2 is...

If the minimum area of the triangle formed by a tangent to the ellipse \({{{x^2}} \over {{b^2}}} + {{{y^2}} \over {4{a^2}}} = 1\) and the co-ordinate axis is kab, then k is equal to _______________.

If x2 + 9y2 \(-\) 4x + 3 = 0, x, y \(\in\) R, then x and y respectively lie in the intervals :

Let L be a common tangent line to the curves 4x2 + 9y2 = 36 and (2x)2 + (2y)2 = 31. Then the square of the slope of the line L is __________.

On the ellipse \({{{x^2}} \over 8} + {{{y^2}} \over 4} = 1\) let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line x + 2y = 0. Let S and S' be the foci of the ellipse and e be its ecc...

Let an ellipse \(E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\), \({a^2} > {b^2}\), passes through \(\left( {\sqrt {{3 \over 2}} ,1} \right)\) and has eccentricity \({1 \over {\sqrt 3 }}\). If a circle, centered at focus F($$\alp...

If a tangent to the ellipse x2 + 4y2 = 4 meets the tangents at the extremities of it major axis at B and C, then the circle with BC as diameter passes through the point :

If the curve x2 + 2y2 = 2 intersects the line x + y = 1 at two points P and Q, then the angle subtended by the line segment PQ at the origin is :

Let \({E_1}:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,a > b\). Let E2 be another ellipse such that it touches the end points of major axis of E1 and the foci of E2 are the end points of minor axis of E1. If E1 and E2 have same e...

Let \(\theta\) be the acute angle between the tangents to the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 1} = 1\) and the circle \({x^2} + {y^2} = 3\) at their point of intersection in the first quadrant. Then tan\(\theta\) is equal to :

Let a tangent be drawn to the ellipse \({{{x^2}} \over {27}} + {y^2} = 1\) at \((3\sqrt 3 \cos \theta ,\sin \theta )\) where \(0 \in \left( {0,{\pi \over 2}} \right)\). Then the value of \(\theta\) such that the sum of intercepts on axes m...

If the points of intersections of the ellipse \({{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1\) and the circle x2 + y2 = 4b, b > 4 lie on the curve y2 = 3x2, then b is equal to :

The length of the minor axis (along y-axis) of
an ellipse in the standard form is \({4 \over {\sqrt 3 }}\). If this
ellipse touches the line, x + 6y = 8; then its
eccentricity is :

Let the line y = mx and the ellipse 2x2 + y2 = 1
intersect at a ponit P in the first quadrant. If the
normal to this ellipse at P meets the co-ordinate axes at \(\left( { - {1 \over {3\sqrt 2 }},0} \right)\) and (0, \(\beta\)), then $$\bet...

If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12,
then the length of its latus rectum is :

If 3x + 4y = 12\(\sqrt 2\) is a tangent to the ellipse
\({{{x^2}} \over {{a^2}}} + {{{y^2}} \over 9} = 1\) for some \(a\) \(\in\) R, then the distance
between the foci of the ellipse is :

Which of the following points lies on the locus of the foot of perpedicular drawn upon any tangent
to the ellipse,
\({{{x^2}} \over 4} + {{{y^2}} \over 2} = 1\)
from any of its foci?

If the normal at an end of a latus rectum of an
ellipse passes through an extremity of the
minor axis, then the eccentricity e of the ellipse
satisfies :

If the co-ordinates of two points A and B are \(\left( {\sqrt 7 ,0} \right)\) and \(\left( { - \sqrt 7 ,0} \right)\) respectively and P is any
point on the conic, 9x2 + 16y2 = 144, then PA + PB is equal to :

Let \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\) (a > b) be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function, \(\phi \left( t \right) = {5 \over {12}} + t - {t^2}\), ...

Let x = 4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is \({1 \over 2}\). If P(1, \(\beta\)), \(\beta\) > 0 is a point on this ellipse, then the equation of the normal to it at P is :

If the tangent to the parabola y2 = x at a point
(\(\alpha\), \(\beta\)), (\(\beta\) > 0) is also a tangent to the ellipse,
x2 + 2y2 = 1, then \(\alpha\) is equal to :

If the tangents on the ellipse 4x2 + y2 = 8 at the
points (1, 2) and (a, b) are perpendicular to each
other, then a2 is equal to :

In an ellipse, with centre at the origin, if the
difference of the lengths of major axis and minor
axis is 10 and one of the foci is at (0,5\(\sqrt 3\)), then
the length of its latus rectum is :

Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If \(\Delta\)S'BS is a right angled triangle with right angle at B and area (\(\Delta\)S'BS) = 8 sq. units, then the length of a latus rectum of...

If the normal to the ellipse 3x2
+ 4y2
= 12 at a point P on it is parallel to the line, 2x + y = 4 and the tangent
to the ellipse at P passes through Q(4,4) then PQ is equal to :

An ellipse, with foci at (0, 2) and (0, –2) and minor axis of length 4, passes through which of the following points?

If tangents are drawn to the ellipse x2 + 2y2 = 2 at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve :

Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following poi...

Let S = \(\left\{ {\left( {x,y} \right) \in {R^2}:{{{y^2}} \over {1 + r}} - {{{x^2}} \over {1 - r}}} \right\};r \ne \pm 1.\) Then S represents :

If the line x – 2y = 12 is tangent to the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\) at the point \(\left( {3, - {9 \over 2}} \right)\) , then the length of the latus
rectum of the ellipse is :