Since the major axis is along the x -axis, the standard equation of the ellipse is given by (x²)/(a²) + (y²)/(b²) = 1, where a is the semi-major axis and b is the semi-minor axis.

Now that we have the values for X and Y, we can find a² and b² by taking the reciprocals.

Substitute the calculated values of a² and b² back into the standard equation of the ellipse to obtain the final equation.

Find the equation of the ellipse, with major axis along the $x$ -axis and passing through the points $(4{,}3)$ and $(-1{,}4)$ .

Answer: $7x^2 + 15y^2 = 247$

Step-by-step solution

Step 1: Standard Equation of Ellipse

Since the major axis is along the $x$ -axis, the standard equation of the ellipse is given by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ , where $a$ is the semi-major axis and $b$ is the semi-minor axis.

Step 2: Substitute the given points into the equation

The ellipse passes through the points $(4{,}3)$ and $(-1{,}4)$ . We substitute these coordinates into the standard equation to form two linear equations in terms of $\frac{1}{a^2}$ and $\frac{1}{b^2}$ .

Step 3: Solve the system of equations

To solve for $a^2$ and $b^2$ , we introduce substitutions $X = \frac{1}{a^2}$ and $Y = \frac{1}{b^2}$ . This transforms the system into two linear equations. We then solve this system using substitution or elimination to find the values of $X$ and $Y$ .

Step 4: Find the values of $a^2$ and $b^2$

Now that we have the values for $X$ and $Y$ , we can find $a^2$ and $b^2$ by taking the reciprocals.

Step 5: Write the equation of the ellipse

Substitute the calculated values of $a^2$ and $b^2$ back into the standard equation of the ellipse to obtain the final equation.

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Find the equation of the ellipse, with major axis along the xxx -axis and passing through the points (4,3)(4{,}3)(4,3) and (−1,4)(-1{,}4)(−1,4).