# How do you find the coordinates of the points on the curve #x^2-xy+y^2=9# where the tangent line is vertical?

Please see the sketch of a solution below.

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Tangents

x^2+y^2-xy-9=0. represents an ellipse.

In the standard form, this is

Substituting in the equation,

graph{((x+y)^2/36+(x-y)^2/18-1)(x-2sqrt3-.35+.01y)(x+2sqrt3+.35+.01y)=0 [-10, 10, -5, 5]}

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To find the coordinates of the points on the curve where the tangent line is vertical, we need to determine the points where the derivative of the curve with respect to x is undefined.

First, we differentiate the equation x^2 - xy + y^2 = 9 with respect to x using implicit differentiation:

d/dx (x^2 - xy + y^2) = d/dx (9) 2x - (x(dy/dx) + y) + 2y(dy/dx) = 0

Next, we solve for dy/dx:

dy/dx = (2x - y) / (x + 2y)

For the tangent line to be vertical, the derivative dy/dx must be undefined. This occurs when the denominator (x + 2y) equals zero.

Setting x + 2y = 0, we can solve for y:

y = -x/2

Substituting this value of y back into the original equation, we get:

x^2 - x(-x/2) + (-x/2)^2 = 9 x^2 + x^2/2 + x^2/4 = 9 (7x^2)/4 = 9 7x^2 = 36 x^2 = 36/7 x = ± √(36/7)

Using the equation y = -x/2, we can find the corresponding y-values:

For x = √(36/7): y = -√(36/7)/2

For x = -√(36/7): y = √(36/7)/2

Therefore, the coordinates of the points on the curve x^2 - xy + y^2 = 9 where the tangent line is vertical are:

(√(36/7), -√(36/7)/2) and (-√(36/7), √(36/7)/2)

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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