Ellipse · Mathematics · JEE Advanced

Consider the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. Let $S(p, q)$ be a point in the first quadrant such that $\frac{p^2}{9}+\frac{q^2}{4}>1$. Two ta...

Consider the ellipse $$$ \frac{x^{2}}{4}+\frac{y^{2}}{3}=1 $$$ Let $H(\alpha, 0), 0 .tg {border-collapse:collapse;border-spacing:0;} .tg td{border...

Let S be the circle in the XY-plane defined the equation x2 + y2 = 4.Let P be a point on the circle S with both coordinates being positive. Let the ta...

Let $${F_1}\left( {{x_1},0} \right)$$ and $${F_2}\left( {{x_2},0} \right)$$ for $${{x_1} < 0}$$ and $${{x_2} > 0}$$, be the foci of the ellipse ...

Let $${F_1}\left( {{x_1},0} \right)$$ and $${F_2}\left( {{x_2},0} \right)$$ for $${{x_1} < 0}$$ and $${{x_2} > 0}$$, be the foci of the ellipse ...

The common tangents to the circle $${x^2} + {y^2} = 2$$ and the parabola $${y^2} = 8x$$ touch the circle at the points $$P, Q$$ and the parabola at th...

The ellipse $${E_1}:{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ is inscribed in a rectangle $$R$$ whose sides are parallel to the coordinate axes. Ano...

Tangents are drawn from the point $$P(3, 4)$$ to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ touching the ellipse at points $$A$$ and $$...

Tangents are drawn from the point $$P(3, 4)$$ to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ touching the ellipse at points $$A$$ and $$...

Tangents are drawn from the point $$P(3, 4)$$ to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ touching the ellipse at points $$A$$ and $$...

The normal at a point $$P$$ on the ellipse $${x^2} + 4{y^2} = 16$$ meets the $$x$$- axis $$Q$$. If $$M$$ is the mid point of the line segment $$PQ$$, ...

Match the conics in Column I with the statements/expressions in Column II : .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:bla...

The line passing through the extremity $$A$$ of the major axis and extremity $$B$$ of the minor axis of the ellipse $${x^2} + 9{y^2} = 9$$ meets its a...

Consider the two curves $${C_1}:{y^2} = 4x,\,{C_2}:{x^2} + {y^2} - 6x + 1 = 0$$. Then,

The minimum area of triangle formed by the tangent to the $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ and coordinate axes is

If tangents are drawn to the ellipse $${x^2} + 2{y^2} = 2,$$ then the locus of the mid-point of the intercept made by the tangents between the coordin...

The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 5} = 1,$$ i...

If $$P=(x, y)$$, $${F_1} = \left( {3,0} \right),\,{F_2} = \left( { - 3,0} \right)$$ and $$16{x^2} + 25{y^2} = 400,$$ then $$P{F_1} + P{F_2}$$ equals

The number of values of $$c$$ such that the straight line $$y=4x + c$$ touches the curve $$\left( {{x^2}/4} \right) + {y^2} = 1$$ is

The radius of the circle passing through the foci of the ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$, and having its centre at $$(0, 3)$$...

The equation $$2{x^2} + 3{y^2} - 8x - 18y + 35 = k$$ represents

Let $$E$$ be the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ and $$C$$ be the circle $${x^2} + {y^2} = 9$$. Let $$P$$ and $$Q$$ be the point...

Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E: \frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P: y^2=12 x$. Suppose that the...

Define the collections {E1, E2, E3, ...} of ellipses and {R1, R2, R3.....} of rectangles as follows :$${E_1}:{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1...

Consider two straight lines, each of which is tangent to both the circle x2 + y2 = (1/2) and the parabola y2 = 4x. Let these lines intersect at the po...

Let $${E_1}$$ and $${E_2}$$ be two ellipses whose centres are at the origin. The major axes of $${E_1}$$ and $${E_2}$$ lie along the $$x$$-axis and th...

An ellipse intersects the hyperbola $$2{x^2} - 2{y^2} = 1$$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If t...

In a triangle $$ABC$$ with fixed base $$BC$$, the vertex $$A$$ moves such that $$$\cos \,B + \cos \,C = 4{\sin ^2}{A \over 2}.$$$ If $$a, b$$ and $$c...

Let $$P\left( {{x_1},{y_1}} \right)$$ and $$Q\left( {{x_2},{y_2}} \right),{y_1} < 0,{y_2} < 0,$$ be the end points of the latus rectum of the el...

On the ellipse $$4{x^2} + 9{y^2} = 1,$$ the points at which the tangents are parallel to the line $$8x = 9y$$ are

Let E be the ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$. For any three distinct points P, Q and Q' on E, let M(P, Q) be the mid-point of...

For how many values of p, the circle x2 + y2 + 2x + 4y $$-$$ p = 0 and the coordinate axes have exactly three common points?

A vertical line passing through the point $$(h,0)$$ intersects the ellipse $${{{x^2}} \over 4} + {{{y^2}} \over 3} = 1$$ at the points $$P$$ and $$Q$$...

Find the equation of the common tangent in $${1^{st}}$$ quadrant to the circle $${x^2} + {y^2} = 16$$ and the ellipse $${{{x^2}} \over {25}} + {{{y^2}...

Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet...

Let $$P$$ be a point on the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,0 < b < a$$. Let the line parallel to $$y$$-axis pas...

Let $$ABC$$ be an equilateral triangle inscribed in the circle $${x^2} + {y^2} = {a^2}$$. Suppose perpendiculars from $$A, B, C$$ to the major axis of...

Find the co-ordinates of all the points $$P$$ on the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, for which the area of the tria...

Consider the family of circles $${x^2} + {y^2} = {r^2},\,\,2 < r < 5$$. If in the first quadrant, the common taingent to a circle of this family...

A tangent to the ellipse x2 + 4y2 = 4 meets the ellipse x2 + 2y2 = 6 at P and Q. Prove that the tangents at P and Q of the ellipse x2 + 2y2 = 6 are at...

Let '$$d$$' be the perpendicular distance from the centre of the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ to the tangent draw...

An ellipse has eccentricity $${1 \over 2}$$ and one focus at the point $$P\left( {{1 \over 2},1} \right)$$. Its one directrix is the common tangent, n...