Find the equation for the ellipse that satisfies the given conditions:Centre at (0,0), major axis on the y-axis and passes through the points(3,2) and (1,6).
Find the equation for the ellipse that satisfies the given conditions:Centre at (0,0), major axis on the y-axis and passes through the points(3,2) and (1,6).
Solution and Explanation
Given that, since the center is at \((0, 0)\) and the major axis is on the \(y-axis\), 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 \((3, 2)\) and \((1, 6)\). Hence,
\(\dfrac{9}{b^2} + \dfrac{4}{a^2}…. (2)\)
\(\dfrac{1}{b^2} + \dfrac{36}{a^2} = 1\)\(…. (3)\)
On solving equations (2) and (3), we obtain \(b^2= 10\) and \(a ^2= 40.\)
Thus, the equation of the ellipse is \(\dfrac{x^2}{10} + \dfrac{y^2}{40} = 1\) or \(4x^2 + y^2 = 40\)
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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}