The area of the quadrilateral formed by the tangents at
the end points of latusrectum to the ellipse is $\frac{x^2}{9}+\frac{y^2}{5}=1$, is
- 27/4 sq units
- 9 sq units
- 27/2 sq uni
- 27 sq units
The Correct Option is D
Approach Solution - 1
To find tangents at the end points of latusrectum , we
find ae
i.e. ae = $\sqrt {a^2-b^2}$ = $ \sqrt4 = 2$
and $\sqrt{b^2(1-e^2)}$ = $\sqrt{5\Big(1-\frac{4}{9}\Big)}$ = $\frac{5}{3}$
By symmetry, the quadrilateral is a rhombus
So, area is four times the area of the right angled
triangle formed by the tangent and axes in the 1st
quadrant.
$\therefore$ Equation of tangent at $\Big(2,\frac{5}{3}\Big)$ is
$\frac{2}{9} x + \frac{5}{3}.\frac{y}{5}$ = 1 $\Rightarrow \frac{3}{\frac{9}{2}}+\frac{y}{3}$ = 1
$therefore$ Area of quadrilateral ABCD
= 4 [area of $\Delta$ AOB]
= 27 sq units
Approach Solution -2
Ans. A quadrilateral, which has four sides, four angles, and four vertices, is a two-dimensional form. A quadrilateral's sides might have equal or unequal lengths based on its characteristics and form. A quadrilateral can have any number of distinct sides, but the total of all these sides is always equal to 360°. The most common shapes for quadrilaterals are parallelograms, squares, rectangles, rhombuses, and trapezoids (trapeziums). A quadrilateral is a figure with four straight lines; the total of all the angles of a quadrilateral is 360°. A quadrilateral can be a parallelogram, square, rectangle, rhombus, trapezium, or kite.
Ellipse is a plane curve that surrounds two focal points, in such a way that for all points on the curve, the sum of two distances to the focal point is a constant. It resembles a circle, which is a special type of ellipse in which both the focal points are the same.
- An ellipse is a closed plane curve, obtained by a point that moves in such a way that the sum of their distances from any two fixed points remains a constant.
- A plane section from a right circular cone is a closed curve.
- The fixed points are called ‘foci’ which are surrounded by the curve.
The equation of the ellipse is given by: x2/a2 + y2/b2 = 1
- The major axis and minor axis define the area of an oval-shaped ellipse.
- The eccentricity that demonstrates the elongation of the ellipse is denoted by the variable “e”.
- Ellipse is part of the conic section. For example, the Parabola and the Hyperbola
- Parabola is open in shape and unbounded. Generally, an ellipse is defined by its equation.
Top Questions on Ellipse
- Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1 \), \( A < B \), have the same eccentricity \( \sqrt{\frac{1}{3}} \). Let the product of their lengths of latus rectums be \( \sqrt{\frac{32}{3}} \) and the distance between the foci of \( E_1 \) be 4. If \( E_1 \) and \( E_2 \) meet at \( A \), \( B \), \( C \), and \( D \), then the area of the quadrilateral ABCD equals:
- Let the product of the focal distances of the point $$ \left( \sqrt{3}, \frac{1}{2} \right) $$ on the ellipse $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad (a > b), $$ be $ \frac{7}{4} $. Then the absolute difference of the eccentricities of two such ellipses is:
- Length of latus rectum of ellipse \[ \frac{x^2}{9} + \frac{y^2}{16} = 1 \]
- Find the center of the ellipse given by the equation \[ 4x^2 + 24x + 9y^2 - 18y + 9 = 0 \]
- If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is:
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The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting, movable piston of area $A$. The left and right compartments are filled with $\frac{3}{2}$ and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant $k$ and natural length $\frac{2L}{5}$. In thermodynamic equilibrium, the piston is at a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $P = \frac{kL}{A} \alpha$, then the value of $\alpha$ is ____
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Let $ S $ denote the locus of the point of intersection of the pair of lines $$ 4x - 3y = 12\alpha,\quad 4\alpha x + 3\alpha y = 12, $$ where $ \alpha $ varies over the set of non-zero real numbers. Let $ T $ be the tangent to $ S $ passing through the points $ (p, 0) $ and $ (0, q) $, $ q > 0 $, and parallel to the line $ 4x - \frac{3}{\sqrt{2}} y = 0 $.
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Concepts Used:
Ellipse
Ellipse Shape
An ellipse is a locus of a point that moves in such a way that its distance from a fixed point (focus) to its perpendicular distance from a fixed straight line (directrix) is constant. i.e. eccentricity(e) which is less than unity
Properties
- Ellipse has two focal points, also called foci.
- The fixed distance is called a directrix.
- The eccentricity of the ellipse lies between 0 to 1. 0≤e<1
- The total sum of each distance from the locus of an ellipse to the two focal points is constant
- Ellipse has one major axis and one minor axis and a center
Read More: Conic Section
Eccentricity of the Ellipse
The ratio of distances from the center of the ellipse from either focus to the semi-major axis of the ellipse is defined as the eccentricity of the ellipse.
The eccentricity of ellipse, e = c/a
Where c is the focal length and a is length of the semi-major axis.
Since c ≤ a the eccentricity is always greater than 1 in the case of an ellipse.
Also,
c2 = a2 – b2
Therefore, eccentricity becomes:
e = √(a2 – b2)/a
e = √[(a2 – b2)/a2] e = √[1-(b2/a2)]
Area of an ellipse
The area of an ellipse = πab, where a is the semi major axis and b is the semi minor axis.
Position of point related to Ellipse
Let the point p(x1, y1) and ellipse
(x2 / a2) + (y2 / b2) = 1
If [(x12 / a2)+ (y12 / b2) − 1)]
= 0 {on the curve}
<0{inside the curve}
>0 {outside the curve}