Find the equation for the ellipse that satisfies the given conditions:Major axis on the x-axis and passes through the points (4,3) and (6,2).
Find the equation for the ellipse that satisfies the given conditions:Major axis on the x-axis and passes through the points (4,3) and (6,2).
Solution and Explanation
Given that :
The major axis is on the x-axis, and the equation of the ellipse will be of the form \(\dfrac{x^2}{b^2} +\dfrac{y^2}{a^2} = 1 ....(1)\), (where ‘\(a\)’ is the semi-major axis.)
The ellipse passes through points \((4, 3)\) and \((6, 2)\).
Hence,
\(\dfrac{16}{a^2} + \dfrac{9}{b^2} = 1…. (2)\)
\(\dfrac{36}{a^2} + \dfrac{4}{b^2} = 1 …. (3)\)
On solving equations (2) and (3), we obtain \(a^2= 5^2\) and \(b^2= 13\)
Thus, the equation of the ellipse is \(\dfrac{x^2}{5^2} +\dfrac{y^2}{13} = 1\)
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:
Questions Asked in CBSE Class XI exam
(i) List the deeds that led Ray Johnson to describe Akhenaten as “wacky”.
(ii) What were the results of the CT scan?
(iii) List the advances in technology that have improved forensic analysis.
(iv) Explain the statement, “King Tut is one of the first mummies to be scanned — in death, as in life...”- CBSE Class XI
- Discovering Tut: The saga Continues
Give reasons for the following.
(i) King Tut’s body has been subjected to repeated scrutiny.
(ii) Howard Carter’s investigation was resented.
(iii) Carter had to chisel away the solidified resins to raise the king’s remains.
(iv) Tut’s body was buried along with gilded treasures.
(v) The boy king changed his name from Tutankhaten to Tutankhamun.- CBSE Class XI
- Discovering Tut: The saga Continues
- Write a note on economic importance of algae and gymnosperms.
- CBSE Class XI
- Bryophytes
- Briefly comment on "The author’s meeting with Norbu."
- CBSE Class XI
- Silk road
- The story revolves around characters who belong to a tribe in Armenia. Mourad and Aram are members of the Garoghlanian family. Now locate Armenia and Assyria on the atlas and prepare a write-up on the Garoghlanian tribes. You may write about people, their names, traits, geographical and economic features as suggested in the story.
- CBSE Class XI
- The summer of the beautiful white horse
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}