Math  /  Calculus

QuestionFind the slope of the tangent line to the ellipse x225+y29=1\frac{x^{2}}{25}+\frac{y^{2}}{9}=1 at the point (x,y)(x, y). slope == \square
Are there any points where the slope is not defined? (Enter them as comma-separated ordered-pairs, e.g., (1,3), ( 2,5)-2,5). Enter none if there are no such points.) slope is undefined at \square

Studdy Solution
Determine if there are any points where the slope is undefined. The slope dydx\frac{dy}{dx} is undefined when the denominator is zero, i.e., when y=0y = 0.
Substitute y=0y = 0 into the ellipse equation:
x225+029=1\frac{x^2}{25} + \frac{0^2}{9} = 1
x225=1\frac{x^2}{25} = 1
x2=25x^2 = 25
x=±5x = \pm 5
The points where the slope is undefined are (5,0)(5, 0) and (5,0)(-5, 0).
The slope of the tangent line is 9x25y-\frac{9x}{25y}. The slope is undefined at (5,0),(5,0)(5, 0), (-5, 0).

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