Finding Equations of Tangent Lines and Normal Lines In Exercises 49 and 50, find equations fur (a) the tangent lines and (b) the normal lines to the hyperbola for the given value of x. (The normal line at a point is perpendicular to the tangent line at the point.)
To-calculate: The tangent lines for the hyperbola at .
The equation of the hyperbola, and the point .
Equation of a line passing through and slope is .
Consider the provided equation of the hyperbola, .
Now, substitute in the equation of the hyperbola, , to find the coordinate for .
thus, the two coordinates are and .
Now, differentiate the equation of the hyperbola, with respect to .
Therefore, is the slope of the curve at any point on the curve of the hyperbola,
So, the slope of the hyperbola, at will be,
Now, consider the standard equation of the line to calculate the equation of the tangent passing through and with a slope
To-calculate: The equations of the normal lines for the hyperbola at .
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