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The equation of the tangent to the curve
y=kx+(6)/(x) at the point
(-2,-19) is
px+qy=c. Find the values of the integers
k,p,q and
c, where
c > 0.

$5$ . The equation of the tangent to the curve $y=k x+\frac{6}{x}$ at the point $(-2,-19)$ is $p x+q y=c$ . Find the values of the integers $k, p, q$ and $c$ , where $c>0$ .

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Find equations of tangent lines using limits

Full solution

Q.

5

. The equation of the tangent to the curve

y=k x+\frac{6}{x}

at the point

(-2,-19)

is

p x+q y=c

. Find the values of the integers

k, p, q

and

c

, where

c>0

.

Find Derivative: First, we need to find the derivative of $y=kx+\frac{6}{x}$ to get the slope of the tangent at the point $(-2,-19)$ . $\newline$ Differentiate $y$ with respect to $x$ : $y' = k - \frac{6}{x^2}$ .
Calculate Slope: Now, plug in the $x$ -coordinate of the point $(-2)$ into the derivative to find the slope of the tangent. $y'(-2) = k - \frac{6}{(-2)^2} = k - \frac{6}{4} = k - 1.5.$
Point-Slope Form: Since the tangent passes through $(-2,-19)$ , we can 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. $\newline$ Substitute $m$ with $k - 1.5$ and $(x_1, y_1)$ with $(-2, -19)$ : $y - (-19) = (k - 1.5)(x - (-2))$ .
Simplify Equation: Simplify the equation: $y + 19 = (k - 1.5)(x + 2)$ . $\newline$ Expand the right side: $y + 19 = kx + 2k - 1.5x - 3$ .
Rearrange Equation: We need to rearrange the equation to the form $px+qy=c$ . Combine like terms: $(k - 1.5)x + y = -19 - 2k + 3$ .
Compare with Standard Form: Now, compare the equation with $px+qy=c$ to find $p$ , $q$ , and $c$ . $p = k - 1.5$ , $q = 1$ , and $c = -19 - 2k + 3$ .
Find $k$ : We know the point $(-2,-19)$ lies on the curve $y=kx+\frac{6}{x}$ , so we can plug it in to find $k$ . $-19 = k(-2) + \frac{6}{-2}$ .
Solve for k: Solve for $k$ : $-19 = -2k - 3$ . Add $2k$ to both sides: $2k - 19 = -3$ . Add $19$ to both sides: $2k = 16$ . Divide by $2$ : $k = 8$ .
Find $p, q, c$ : Now that we have $k$ , we can find $p, q,$ and $c$ . $p = k - 1.5 = 8 - 1.5 = 6.5$ , but $p$ must be an integer, so there's a mistake.

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