Chapter 9.4, Problem 6OE

To determine

To calculate: To find the x- and y-intercepts of the ellipse and to tell on which of the coordinate axes its foci lie

Answer to Problem 6OE

x-intercept is $\sqrt{3}$ and y-intercept is $\sqrt{2}$ and foci lies on x-axis.

Explanation of Solution

Given information: Ellipse is $2x^2+3y^2=6$

Formula Used:

Standard form of the equation of ellipse with center $\left(h,k\right)$ and major axis parallel to x-axis is:

  $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1$

Where, $a=$ x-intercept, $b=$ y-intercept

When, $a>b$ , foci lies on x-axis

When $b>a$ , foci lies on y-axis

Calculation:

Equation of ellipse is given as follows:

  $2x^2+3y^2=6$

Divide both sides of equation by $6$ :

  $\begin{array}{l}\frac{x^2}{3}+\frac{y^2}{2}=1\\ \frac{x^2}{{\left(\sqrt{3}\right)}^2}+\frac{y^2}{{\left(\sqrt{2}\right)}^2}=1\end{array}$

Comparing the equation with standard form of the equation of ellipse $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1$ :

  $\begin{array}{l}a=\sqrt{3}\\ b=\sqrt{2}\end{array}$

Since $a>b$ , thus foci lies on x-axis.

Conclusion:

Hence, x-intercept is $\sqrt{3}$ and y-intercept is $\sqrt{2}$ and foci lies on x-axis.