The given equation is \( \dfrac{x^2}{36} + \dfrac{y^2}{16} = 1\)
Here, the denominator of \(\dfrac{x^2}{36}\) is greater than the denominator of \(\dfrac{y^2}{16}\). Therefore, the major axis is along the \(x-axis\), while the minor axis is along the \(y-axis.\)
On comparing the given equation with \( \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) we obtain \(a = 6\) and \(b = 4.\)
\(∴ c = √(a^2 – b^2)\)
\(= √(36-16)\)
\(= √20\)
\(= 2√5\)
Therefore, the coordinates of the foci are \((2√5, 0)\) and \((-2√5, 0).\)
The coordinates of the vertices are \((6, 0)\) and \((-6, 0).\)
Length of major axis =\( 2a= 12\)
Length of minor axis = \(2b= 8\)
Eccentricity, \(e = \dfrac{c}{a} = \dfrac{2√5}{6} = \dfrac{√5}{3}\)
Length of latus rectum = \(\dfrac{2b^2}{a} = \dfrac{(2×16)}{6} = \dfrac{16}{3}.\) (Ans)
Figures 9.20(a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect ? Why ?
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
Read More: Conic Section
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)]
The area of an ellipse = πab, where a is the semi major axis and b is the semi minor axis.
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}