Ellipse

Consider the ellipses given by
\(x^2+4 y^2=1 \quad \text { and } \quad 4 x^2+y^2=1\)
Let $P$ be the point in the first quadrant where the given ellipses intersect. If $\theta$ is the acute angle between the tangents to the given ellipses ...

Let $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right)$ be two distinct points on the ellipse
\(\frac{x^2}{9}+\frac{y^2}{4}=1\)
such that $y_1>0$, and $y_2>0$. Let $C$ denote the circle $x^2+y^2=9$, and $M$ be the point $(3,0)$.
Suppose...

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 tangents are drawn from $S$ to the ellipse, of which one meets the ellipse at one end 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 tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$, respectively and the tan...

Consider the ellipse

\($ \frac{x^{2}}{4}+\frac{y^{2}}{3}=1\)$

Let $H(\alpha, 0), 0<\alpha<2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respe...

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 the line segment joining P and Q, and M(P, Q') be the mid-point of the line segment joini...

Define the collections {E1, E2, E3, ...} of ellipses and {R1, R2, R3.....} of rectangles as follows :\({E_1}:{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1\)R1 : rectangle of largest area, with sides parallel to the axes, inscribed in E1;En : el...

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 point Q. Consider the ellipse whose centre is at the origin O(0, 0) and whose semi-major axi...

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 tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point o...

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

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 \({{{x^2}} \over 9} + {{{y^2}} \over 8} = 1\). Suppose a parabola having vertex at the origin an...

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 \({{{x^2}} \over 9} + {{{y^2}} \over 8} = 1\). Suppose a parabola having vertex at the origin an...

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 the \(y\)-axis, respectively. Let \(S\) be the circle $${x^2} + {\left( {y - 1} \right)^2} ...

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 the points \(R\), \(S\). Then the area of the quadrilateral \(PQRS\) is

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 \(Q\). Let the tangents to the ellipse at \(P\) and \(Q\) meet at the point \(R\). If $$\Delta...

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. Another ellipse \({E_2}\) passing through the point \((0, 4)\) circumscribes the rectangle $$...

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 \(B\).
The equation of the locus of the point whose distances from the point \(P\) and the l...

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 \(B\).
The orthocentre of the triangle \(PAB\) is

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 \(B\).
The coordinates of \(A\) and \(B\) are

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 the axes of the ellipse are along the coordinate axes then

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 \(PQ\), then the locus of \(M\) intersects the latus rectums of the given ellipse at the points

Match the conics in Column I with the statements/expressions in Column II :

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ove...

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 auxiliary circle at the point \(M\). Then the area of the triangle with vertices at \(A\), ...

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 \(c\) denote the lengths of the sides of the triangle opposite to the angles \(A, B\) and $$C...

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

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 ellipse \({x^2} + 4{y^2} = 4.\) The equations of parabolas with latus rectum \(PQ\) are :

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

Find the equation of the common tangent in the first quadrant to the circle \(x^{2}+y^{2}=16\) and the ellipse \(\frac{x^{2}}{25}+\frac{y^{2}}{4}=1\). Also find the length of the intercept of the tangent between the coordinate axes.

Find the equation of the common tangent in \({1^{st}}\) quadrant to the circle \({x^2} + {y^2} = 16\) and the ellipse \({{{x^2}} \over {25}} + {{{y^2}} \over 4} = 1\). Also find the length of the intercept of the tangent between the coordin...

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 coordinate axes is

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

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 on the corresponding directrix.

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 passing through \(P\) meet the circle \({x^2} + {y^2} = {a^2}\) at the point \(Q\) such that \(P\) ...

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 the ellipse \(x.{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\), \((a>b)\) meets ...

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 triangle \(PON\) is maximum, where \(O\) denotes the origin and \(N\), the foot of the perpend...

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

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 and the ellipse \(4{x^2} + 25{y^2} = 100\) meets the co-ordinate axes at \(A\) and \(B\), then ...

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

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 right angles.

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, nearer to the point \(P\), to the circle \({x^2} + {y^2} = 1\) and the hyperbol;a $${x^2} -...

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 \((0, 3)\) is

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 drawn at a point \(P\) on the ellipse. If \({F_1}\) and \({F_2}\) are the two foci of the elli...

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 points \((1, 2)\) and \((2, 1)\) respectively. Then