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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

STEP 1

1. The ellipse is given by the equation x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1.
2. We need to find the slope of the tangent line at a general point (x,y)(x, y) on the ellipse.
3. The slope of the tangent line can be found using implicit differentiation.
4. We will also determine if there are points where the slope is undefined.

STEP 2

1. Differentiate the equation of the ellipse implicitly with respect to xx.
2. Solve for dydx\frac{dy}{dx} to find the slope of the tangent line.
3. Determine if there are any points where the slope is undefined.

STEP 3

Differentiate the equation of the ellipse x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1 implicitly with respect to xx.
ddx(x225)+ddx(y29)=ddx(1)\frac{d}{dx}\left(\frac{x^2}{25}\right) + \frac{d}{dx}\left(\frac{y^2}{9}\right) = \frac{d}{dx}(1)
Using the chain rule, we differentiate each term:
2x25+2y9⋅dydx=0\frac{2x}{25} + \frac{2y}{9} \cdot \frac{dy}{dx} = 0

STEP 4

Solve for dydx\frac{dy}{dx} to find the slope of the tangent line.
2y9⋅dydx=−2x25\frac{2y}{9} \cdot \frac{dy}{dx} = -\frac{2x}{25}
dydx=−2x25⋅92y\frac{dy}{dx} = -\frac{2x}{25} \cdot \frac{9}{2y}
Simplify the expression:
dydx=−9x25y\frac{dy}{dx} = -\frac{9x}{25y}
The slope of the tangent line at the point (x,y)(x, y) is −9x25y-\frac{9x}{25y}.

STEP 5

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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