Ellipse and Hyperbola · Mathematics · WB JEE

The total number of tangents through the point (3, 5) that can be drawn to the ellipse 3x2 + 5y2 = 32 and 25x2 + 9y2 = 450 is...

The line y = 2t2 intersects the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ in real point if

The angle between the line joining the foci of an ellipse to one particular extremity of the minor axis is 90$$^\circ$$. The eccentricity of the ellip...

If the line ax + by + c = 0 is a tangent to the curve xy = 4, then

S and T are the foci of an ellipse and B is an end point of the minor axis. If STB is an equilateral triangle, the eccentricity of the ellipse is...

For different values of $$\alpha$$, the locus of the point of intersection of the two straight lines $$\sqrt 3 x - y - 4\sqrt 3 \alpha = 0$$ and $$\s...

The eccentricity of the hyperbola $$4{x^2} - 9{y^2} = 36$$ is

The length of the latus rectum of the ellipse is 16x2 + 25y2 = 400 is

The equation 8x2 + 12y2 $$-$$ 4x + 4y $$-$$ 1 = 0 represents

If a hyperbola passes through the point P($$\sqrt2$$, $$\sqrt3$$) and has foci at ($$\pm$$ 2, 0), then the tangent to this hyperbola at P is

The average length of all vertical chords of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1,a \le x \le 2a$$, is :

The tangent at point $$(a\cos \theta ,b\sin \theta ),0 ...

Let $$A(2\sec \theta ,3\tan \theta )$$ and $$B(2\sec \phi ,3\tan \phi )$$ where $$\theta + \phi = {\pi \over 2}$$ be two points on the hyperbola $$...

If the lines joining the focii of the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ where $$a > b$$, and an extremity of its minor...

Let $$P(3\sec \theta ,2\tan \theta )$$ and $$Q(3\sec \phi ,2\tan \phi )$$ be two points on $${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$$ such that $$\...

AB is a variable chord of the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$. If AB subtends a right angle at the origin O, then $$...

PQ is a double ordinate of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ such that $$\Delta OPQ$$ is an equilateral triangle...

The normal to a curve at P(x, y) meets the X-axis at G. If the distance of G from the origin is twice the abscissa of P then the curve is

The co-ordinate of a point on the auxiliary circle of the ellipse x2 + 2y2 = 4 corresponding to the point on the ellipse whose eccentric angle is 60$$...

The locus of the centre of a variable circle which always touches two given circles externally is

The points of intersection of two ellipses $${x^2} + 2{y^2} - 6x - 12y + 20 = 0$$ and $$2{x^2} + {y^2} - 10x - 6y + 15 = 0$$ lie on a circle. The cent...

A double ordinate PQ of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ is such that $$\Delta OPQ$$ is equilateral, O being th...

If B and B' are the ends of minor axis and S and S' are the foci of the ellipse $${{{x^2}} \over {25}} + {{{y^2}} \over 9} = 1$$, then the area of the...

Consider the curve $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$. The portion of the tangent at any point of the curve intercepted between...

Consider the curve $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$. The portion of the tangent at any point of the curve intercepted between...

Let P(4, 3) be a point on the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$. If the normal at P intersects the X-axis at (16, 0)...

For the hyperbola $${{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1$$, which of the following remains fixed when $$\al...

S and T are the foci of an ellipse and B is the end point of the minor axis. If STB is equilateral triangle, the eccentricity of the ellipse is

The equation of the directrices of the hyperbola $$3{x^2} - 3{y^2} - 18x + 12y + 2 = 0$$ is

P is the extremity of the latusrectum of ellipse $$3{x^2} + 4{y^2} = 48$$ in the first quadrant. The eccentric angle of P is

The length of conjugate axis of a hyperbola is greater than the length of transverse axis. Then, the eccentricity e is

A point is in motion along a hyperbola $$y = {{10} \over x}$$ so that its abscissa x increases uniformly at a rate of 1 unit per second. Then, the rat...

Let the eccentricity of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ be reciprocal to that of the ellipse x2 + 9y2 = 9, the...

Let P be a point on the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ and the line through P parallel to the Y-axis meets the circle x2 + y2 =...

Let P be the foot of the perpendicular from focus S of hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ on the line bx $$-$$ ay = 0...

B is an extremity of the minor axis of an ellipse whose foci are S and S'. If $$\angle SBS'$$ is a right angle, then the eccentricity of the ellipse i...

The line segment joining the foci of the hyperbola $${x^2} - {y^2} + 1 = 0$$ is one of the diameters of a circle. The equation of the circle is

Let A($$-$$ 1, 0) and B(2, 0) be two points. A point M moves in the plane in such a way that $$\angle MBA$$ = 2$$\angle MAB$$. Then, the point M moves...

Tangents are drawn to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 5} = 1$$ at the ends of both latusrectum. The area of the quadrilateral, so fo...

Let f be a strictly decreasing function defined on R such that $$f(x) > 0,\forall x \in R$$. Let $${{{x^2}} \over {f({a^2} + 5a + 3)}} + {{{y^2}} \ove...

Chords of an ellipse are drawn through the positive end of the minor axis. Their midpoint lies on

The line y = x + 5 touches

Consider a tangent to the ellipse $${{{x^2}} \over 2} + {{{y^2}} \over 1} = 1$$ at any point. The locus of the mid-point of the portion intercepted be...

Equation of a tangent to the hyperbola 5x2 $$-$$ y2 = 5 and which passes through an external point (2, 8) is

A hyperbola, having the transverse axis of length 2sin$$\theta$$ is confocal wit6h the ellipse 3x2 + 4y2 = 12. Its equation is