Find the equation of the ellipse, with major axis along the x -axis and passing through the points (4,3) and (−1,4).
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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 a2x2+b2y2=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 a21 and b21.
Step 3: Solve the system of equations
To solve for a2 and b2, we introduce substitutions X=a21 and Y=b21. 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 a2 and b2
Now that we have the values for X and Y, we can find a2 and b2 by taking the reciprocals.
Step 5: Write the equation of the ellipse
Substitute the calculated values of a2 and b2 back into the standard equation of the ellipse to obtain the final equation.