Coordinate geometry question..help me

The difference between the distance of point P from the point (4,0) and the distance of Pfrom the point (-4,0) is 2(surd10) units.Show that the locus of P is the hyperbola x^2/10-y^2/6=1

Show that the line y=x-2 is a tangent to this curve

thanks for helping
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If P is (x,y) distance from (4,0) = sqrt[(x-4)^2+y^2]
Distance from (-4,0) =sqrt[(x+4)^2+y^2]
Given sqrt[(x+4)^2+y^2]-sqrt[(x-4)^2+y^2]=2s (where s=root10)
Gives sqrt[(x+4)^2+y^2]=2s-sqrt[(x-4)^2+y^2]
Square both sides: (x+4)^2+y^2 = 40 -4ssqrt[(x-4)^2+y^2]+(x-4)^2+y^2
so 4ssqrt[(x-4)^2+y^2]=40+(x-4)^2+y^2-(x+4)…
Divide by 4
ssqrt[(x-4)^2+y^2]=10-4x
Square both sides
10[x^2-8x+16+y^2]=100-80x+16^x^2
6x^2 - 10y^2=60 and divide each term by 60 to give the answer

You can do the second part by finding the equation in x satisfied by the poin(s) where
the line meets the hyperbola.So sub y=x-2 to give 6x^2-10(x-2)^2=60
Clean up and show that the equation has only one real root ( two equal roots)which means
that the line is a tangent.