Question

If $\dfrac{{{x^2}}}{{8 - a}} + \dfrac{{{y^2}}}{{a - 2}} = 1$ represents an ellipse. Then find the range of $'\;a'$ .

Answer 1

To solve these questions compare the given equation of the ellipse to the general equation of an ellipse, that is $\dfrac{{{x^2}}}{{{a^2}}} + \dfrac{{{y^2}}}{{{b^2}}} = 1$ where ${a^2} > 0,{b^2} > 0$ and the major and the minor axis of the ellipse are represented by $a$ and $b$ respectively. After comparing the terms of the two equations apply inequality conditions as given in the question.

Complete step-by-step solution:
The general equation for an ellipse is $\dfrac{{{x^2}}}{{{a^2}}} + \dfrac{{{y^2}}}{{{b^2}}} = 1$ , where ${a^2} > 0$ and ${b^2} > 0$ . The major and minor axis of the ellipse is denoted by $a$ and $b$ . This equation also represents that the center of the ellipse is at the origin and the major axis is along the $x$ - axis.
Given in the question $\dfrac{{{x^2}}}{{8 - a}} + \dfrac{{{y^2}}}{{a - 2}} = 1$ ;
Comparing the above equation and the general equation of an ellipse along with the conditions that are given with the general form, we get
$\Rightarrow 8 - a > 0$ And $a - 2 > 0$
In first inequality add $a$ to both sides and in the second inequality add $2$ on both sides of the inequality, to get
$a < 8$ And $a > 2$

Which implies that the range of $'\;a'$ for the above question is $a \in \left( {2,8} \right)$ , that is the value of $a$ lies in between $2$ and $8$ , excluding both the endpoints.

Additional Information: An ellipse can be defined as the locus of all points in a plane, the sum of whose distances from the two fixed points on the plane is constant. The fixed points are the foci of the ellipse. An ellipse consists of a major axis and a minor axis. Ellipses are very common in physics, astronomy, and engineering.

Note: While solving questions on ellipses the most common mistake that can occur is exchanging the variables for the major and minor axis. Also, note carefully whether the length of the axis or the radius is given in the question. Another common error to avoid will be to note that the denominators in the equation of the ellipse contain the squares of the radii, and not the radii themselves.