How do you find the equations for the normal line to #x^2+y^2=9# through (0,3)?
We find the derivative.
We now determine the slope of the tangent.
Hopefully this helps!
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Here is a solution using geometry.
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Here is a solution using calculus, but without implicit differentiation.
Differentiate using the chain rule to get
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To find the equation of the normal line to the circle (x^2 + y^2 = 9) through the point ((0,3)), follow these steps:
 First, find the derivative of the circle's equation.
 Then, find the slope of the tangent line at the point of tangency (which is the point of intersection between the circle and the normal line).
 Use the negative reciprocal of this slope to find the slope of the normal line.
 Finally, use the pointslope form to write the equation of the normal line.
Here are the steps in more detail:

The equation of the circle is (x^2 + y^2 = 9). Taking the derivative with respect to (x), we get: [2x + 2y\frac{{dy}}{{dx}} = 0] [y\frac{{dy}}{{dx}} = x] [ \frac{{dy}}{{dx}} = \frac{{x}}{{y}}]

At the point ((0,3)), substitute (x = 0) and (y = 3) into the derivative to find the slope of the tangent line: [m_{\text{tangent}} = \frac{{0}}{{3}} = 0]

The slope of the normal line is the negative reciprocal of the slope of the tangent line: [m_{\text{normal}} = \frac{1}{{m_{\text{tangent}}}} = \text{undefined}]

Since the slope of the normal line is undefined, the normal line is vertical and passes through the point ((0,3)). Therefore, its equation is (x = 0).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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