To show: The tangent line to the ellipse at the point is .
The equation of ellipse is .
The equation of ellipse at is .
Derivative rule: Chain rule
If and are both differentiable function, then .
The equation of the tangent line at is, (1)
Where, m is the slope of the tangent line at and .
Obtain the equation of tangent line to the ellipse at .
Consider the equation of ellipse .
Differentiate implicitly with respect to x,
Apply the chain rule (1) and simplify the terms,
Multiply the equation by ,
Thus, the derivative of the equation is .
The slope of the tangent line at is computed as follows,
Substitute for in ,
Therefore, the slope of the tangent at is .
Substitute for and in equation (1),
Divided the equation by on both sides,
Substitute in the above equation,
Hence, it can be concluded that the tangent line to the ellipse at the point is .
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