Math / Geometry

QuestionFind the equation of the tangentizine at the given poin 7. $x^{2}-y^{2}=27$ at $(6,-3)$

Studdy Solution

STEP 1

1. The given equation is $x^2 - y^2 = 27$ .
2. We need to find the equation of the tangent line at the point $(6, -3)$ .

STEP 2

1. Differentiate the given equation implicitly to find the derivative.
2. Evaluate the derivative at the given point to find the slope of the tangent line.
3. Use the point-slope form to find the equation of the tangent line.

STEP 3

Differentiate the given equation implicitly with respect to $x$ :
$\frac{d}{dx}(x^2 - y^2) = \frac{d}{dx}(27)$
$2x - 2y \frac{dy}{dx} = 0$

STEP 4

Solve for $\frac{dy}{dx}$ to find the slope of the tangent line:
$2x - 2y \frac{dy}{dx} = 0$
$2y \frac{dy}{dx} = 2x$
$\frac{dy}{dx} = \frac{x}{y}$
Evaluate the derivative at the point $(6, -3)$ :
$\frac{dy}{dx} \bigg|_{(6, -3)} = \frac{6}{-3} = -2$

STEP 5

Use the point-slope form of the equation of a line:
$y - y_1 = m(x - x_1)$
where $m$ is the slope and $(x_1, y_1)$ is the point $(6, -3)$ .
Substitute the values:
$y - (-3) = -2(x - 6)$
$y + 3 = -2x + 12$
Rearrange to find the equation of the tangent line:
$y = -2x + 9$
The equation of the tangent line is:
$\boxed{y = -2x + 9}$

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