#39: for the graph of the funtion y=1/x^2 a tangent formed through (a,1/a^2)
A. Express using "a" the equation of the tangent
B. Express using "a" the unions with the coordinate systems (x=0),(y=0)
C. find "a" for whom the tangent creates "Isosceles triangle" with the coordinate systems
A. dy/dx (1/x^2) = -2/x^3
Equation of the tangent at "a"
y - 1/ a^2 = (-2/a^3)(x - a)
y - 1/a^2 = -2x/a^3 + 2/a^2
y = -2x/a^3 + 3/a^2
y = [3a - 2x] / a^3
B. y intercept → x = 0 → y = 3/a^2 → (0, 3/a^2 )
x intercept → y = 0 → x = 3a/2 → ( 3a/2 , 0 )
C. Notice that many values of 'a' will produce isosceles triangles.......for instance......it we specify that the length of both tangent lines between the x and y axis = 5, we have
9/a^4 + 9a^2/4 = 25
9 + (9/4)a^6 = 25a^4
(9/4)a^6 - 25a^4 + 9 = 0
a≈ .78575 and a ≈ -.78575
Look at the graph, here......https://www.desmos.com/calculator/0twc2fqneu
Notice that two sides of the triangle formed by the points of the intersection of the tangent lines with both the x and y axis have lengths ≈ 5
