Find the equation for the ellipse that satisfies the given conditions: Foci \((±3,0),\) \(a=5\)
Solution and Explanation
Gven that, foci \(( ±3, 0)\), \(a= 4 \)
Since the foci are on the \(x-axis\), the major axis is along the \(x-axis\)
Therefore, the equation of the ellipse will be of the form \(\dfrac{x^2}{a^2} +\dfrac{ y^2}{b^2} = 1\), where \(a\) is the semi-major axis.
Accordingly, \(c = 3 \) and \(a = 4\).
It is known that
\(a^2 = b^2 + c^2\)
\(∴ 4^2 = b^2 + 3^2\)
\(⇒ 16 = b^2 + 9\)
\(⇒ b^2 = 16 – 9\)
\(⇒ b^2 = 7\)
Thus, the equation of the ellipse is \(\dfrac{x^2}{16} + \dfrac{y^2}{7} = 1\)
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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}