Find the equation of the tangent line to the circle at a given point $(3, 4)$ if the equation of the circle is $x^2 + y^2 = 25$
Answer 1
Ava Martin
To find the equation of the tangent line to the circle at the point $(3, 4)$, follow these steps:
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The equation of the circle is:
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$ x^2 + y^2 = 25 $
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The gradient of the radius at the point $(3, 4)$ is:
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$ \x0crac{4 – 0}{3 – 0} = \x0crac{4}{3} $
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The gradient of the tangent line at $(3, 4)$ is the negative reciprocal of the gradient of the radius:
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$ \text{Gradient of tangent} = -\x0crac{3}{4} $
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Using the point-slope form equation of a line:
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$ y – y_1 = m(x – x_1) $
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where $(x_1, y_1) = (3, 4)$ and $m = -\x0crac{3}{4}$:
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$ y – 4 = -\x0crac{3}{4}(x – 3) $
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Expanding and simplifying:
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$ 4(y – 4) = -3(x – 3) $
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$ 4y – 16 = -3x + 9 $
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$ 4y + 3x = 25 $
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Thus, the equation of the tangent line is:
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$ 4y + 3x = 25 $
Answer 2
Matthew Carter
To find the equation of the tangent line to the circle at the point $(3, 4)$, follow these steps:
The equation of the circle is:
$ x^2 + y^2 = 25 $
The gradient of the radius at the point $(3, 4)$ is:
$ x0crac{4}{3} $
The gradient of the tangent line at $(3, 4)$ is:
$ -x0crac{3}{4} $
Using the point-slope form:
$ y – 4 = -x0crac{3}{4}(x – 3) $
Expanding and simplifying:
$ 4y – 16 = -3x + 9 $
$ 4y + 3x = 25 $
The tangent line equation is:
$ 4y + 3x = 25 $
Answer 3
Charlotte Davis
The equation of the tangent line to the circle at the point $(3, 4)$ is:
$ 4y + 3x = 25 $
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