#x-y=1=># write in slope-intercept form:

#y=x-1=># slope = 1

#5x^2+ 6xy +5y^2 =8=># implicit differentiation:

#10x+6y+6xy'+10yy'=0#

#y'(3x+5y)=-(5x+3y)#

#y'=-(5x+3y)/(3x+5y)=># the slope of the curve, equate to 1:

#-(5x+3y)/(3x+5y)=1=># solve for #y# in terms of #x#:

#3x+5y=-5x-3y#

#8x+8y=0#

#x+y=0#

#y=-x=># substitute in the equation of the curve, solve for #x#:

#5x^2+6x*(-x)+5*(-x)^2=8#

#5x^2-6x^2+5x^2=8#

#4x^2=8#

#x^2=2#

#x=-sqrt2 , sqrt2=># solve for #y:#

#y=sqrt2 , -sqrt2#

hence the required points are:

#(x, y)=(-sqrt2, sqrt2) , (sqrt2, -sqrt2)#