If P is at (-2,3) and Q is at (6,-3),the line segment [PQ] is a tangent to y=b/(x+1)^2 Find b.

Question

If P is at (-2,3) and Q is at (6,-3),the line segment [PQ] is a tangent to y=b/(x+1)^2 Find b.

 

please explain full

 

Answer 1

Slope of tangent line = (-3-3)/(6-(-2)) = -3/4

 

Equation of tangent line:  y - 3 = (-3/4)(x - (-2))

 

                                    y - 3 = (-3/4)x - 3/2

 

                                         y = (-3/4)x + 3/2

 

At the point of tangency (the point where the tangent line intersects the curve), the y-coordinate of the point on the tangent line is the same as the y-coordinate of the point on the graph of y = b/(x+1)².

 

So, the x-coordinate of the point of tangency must satisfy the equation (-3/4)x + 3/2 = b/(x+1)²  (*)

 

From Calculus, the slope of the tangent line to a curve is the derivative evaluated at the x-coordinate of the point of tangency.

 

So, y' = -2b/(x+1)³ = -3/4

 

Solving for b/(x+1)², we have b/(x+1)² = (3/4)[(x+1)/2]

 

Simplify to get b/(x+1)² = (3/8)(x+1)  (**)

 

Using equations (*) and (**), we now have two different expressions for b/(x+1)².

 

Equating these expressions, (3/8)(x+1) = (-3/4)x + 3/2

 

                                        (3/8)x + 3/8 = (-3/4)x + 3/2

 

Multiply by 8 to obtain 3x + 3 = -6x + 12

 

                                      9x = 9

 

So, x = 1

 

Finally, since x = 1 and b/(x+1)² = (3/8)(x+1), we have b/4 = 3/4.

 

Therefore b = 3.   

 

 

 

 

Answer 2

Tangent is possible  on points P and Q .

 

If P is the tangent point. 3=b/1² which makes b=1

 

If Q is the tangent point. -3=b/7² which makes b=-147

These values simple prove the possibility of  points P and Q on curve .

 

Slope of PQ is =-3/4

 

Derivative of y=-2b/(x+1)³

 

If P is the tangent point. -3/4=-2b/-1³ which makes b=-3/8 ( values not matching.

 

 

If Q is the tangent point. -3/4=-2b/7³ which makes b=-3/8 ( value is not matching.)

 

 

 

If Q is the tangent point. -3/4=-2b/7³ which makes b=347/8 ( value is not matching.)

 

Data is wrong!!!