Question:

The equation $5x^2 + y^2 + y = 8$ represents

Updated On: Jun 8, 2024
  • an ellipse
  • a parabola
  • a hyperbola
  • a circle
Hide Solution
collegedunia
Verified By Collegedunia

The Correct Option is A

Solution and Explanation

We have,
$5 x^{2}+y^{2}+y=8$
$\Rightarrow 5 x^{2}+(y+1 / 2)^{2}-\left(\frac{1}{2}\right)^{2}=8$
$\Rightarrow 5 x^{2}+\left(y+\frac{1}{2}\right)^{2}=\frac{33}{8}$
$\Rightarrow \frac{5 x^{2}}{\frac{33}{8}}+\frac{(y+1 / 2)^{2}}{\frac{33}{8}}=1$
$\Rightarrow \frac{x^{2}}{\left(\frac{33}{40}\right)}+\frac{(y+1 / 2)^{2}}{\left(\frac{33}{8}\right)}=1$
which is an equation of ellipse.
Was this answer helpful?
3
0

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}