The Y-intercept of the common tangent to the parabola y^2=32 x and x^2=108 y is

Step-by-step solution

A common tangent to both parabolas must have the same slope m and y-intercept c in its equation y = mx + c . For y^2 = 32x , which has standard form y^2 = 4ax with 4a = 32 giving a = 8 , the tangent with slope m is y = mx + 8 m , yielding y-intercept 8 m . For x^2 = 108y , in standard form x^2 = 4by with 4b = 108 giving b = 27 , the tangent with slope m is y = mx - 27m^2 , yielding y-intercept -27m^2 . Equating the y-intercepts for a common tangent: 8 m = -27m^2 Multiplying both sides by m gives 8 = -27m^3 , so m^3