find equations of all lines having slope - 3 that are tangent to the curve ( y=\frac{12}{x - 5} ).\nselect…

find equations of all lines having slope - 3 that are tangent to the curve ( y=\frac{12}{x - 5} ).\nselect the correct choice below and fill in the answer box(es) within your choice.\na. there are two lines tangent to the curve with a slope of - 3. the equation of the line with the larger y - intercept is ( square ) and the equation of the line with the smaller y - intercept is ( square ).\n(type equations.)\nb. there is only one line tangent to the curve with a slope of - 3 and its equation is ( square ).\n(type an equation.)

find equations of all lines having slope - 3 that are tangent to the curve ( y=\frac{12}{x - 5} ).\nselect the correct choice below and fill in the answer box(es) within your choice.\na. there are two lines tangent to the curve with a slope of - 3. the equation of the line with the larger y - intercept is ( square ) and the equation of the line with the smaller y - intercept is ( square ).\n(type equations.)\nb. there is only one line tangent to the curve with a slope of - 3 and its equation is ( square ).\n(type an equation.)

Answer

Explanation:

Step1: Find the derivative of the function

The function is (y = \frac{12}{x - 5}=(x - 5)^{-12}). Using the power rule ((u^n)^\prime=nu^{n - 1}u^\prime), where (u=x - 5), (n=- 12) and (u^\prime = 1). The derivative (y^\prime=-\frac{12}{(x - 5)^2}).

Step2: Set the derivative equal to the slope

We want the slope (y^\prime=-3). So, (-\frac{12}{(x - 5)^2}=-3). Cross - multiply gives (12 = 3(x - 5)^2). Then ((x - 5)^2 = 4). Taking square roots, (x-5=\pm2).

  • Case 1: (x-5 = 2), then (x = 7)
  • Case 2: (x-5=-2), then (x = 3)

Step3: Find the corresponding (y) - values

  • When (x = 7), (y=\frac{12}{7 - 5}=6)
  • When (x = 3), (y=\frac{12}{3 - 5}=-6)

Step4: Use the point - slope form (y - y_1=m(x - x_1))

  • For the point ((7,6)) and (m=-3), (y-6=-3(x - 7)), which simplifies to (y=-3x + 27)
  • For the point ((3,-6)) and (m=-3), (y+6=-3(x - 3)), which simplifies to (y=-3x+3)

Answer:

A. There are two lines tangent to the curve with a slope of (-3). The equation of the line with the larger (y) - intercept is (y=-3x + 27) and the equation of the line with the smaller (y) - intercept is (y=-3x+3)