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 $ e_1 $ and $ e_2 $ be the eccentricities of the ellipse $ \frac{x^2}{b^2} + \frac{y^2}{25} = 1 $ and the hyperbola $ \frac{x^2}{16} - \frac{y^2}{b^2} = 1, $ respectively. If $ b<5 $ and $ e_1 e_2 = 1 $, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is:
- If the equation of an ellipse \( E \) is \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \), then the length of the latus rectum of \( E \) is?
- 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:
- 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 length of a latus rectum of an ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ be 10. If its eccentricity is $ e $, and the minimum value of the function $ f(t) = t^2 + t + \frac{11}{12} $, where $ t \in \mathbb{R} $, then $ a^2 + b^2 $ is equal to:
Questions Asked in JEE Advanced exam
- Consider a star of mass $ m_2 $ kg revolving in a circular orbit around another star of mass $ m_1 $ kg with $ m_1 \gg m_2 $. The heavier star slowly acquires mass from the lighter star at a constant rate of $ \gamma $ kg/s. In this transfer process, there is no other loss of mass. If the separation between the centers of the stars is $ r $, then its relative rate of change $ \frac{1}{r} \frac{dr}{dt} $ (in s$^{-1}$) is given by:
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Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is
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As shown in the figures, a uniform rod $ OO' $ of length $ l $ is hinged at the point $ O $ and held in place vertically between two walls using two massless springs of the same spring constant. The springs are connected at the midpoint and at the top-end $ (O') $ of the rod, as shown in Fig. 1, and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is $ f_1 $. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2, and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is $ f_2 $. Ignoring gravity and assuming motion only in the plane of the diagram, the value of $\frac{f_1}{f_2}$ is:
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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}