What is the direction of the line passing through the point (1,-2) having the maximum slope?

the complete question in the attached figure

a) gradient vector for this function 

f(x,y)-------- > 2x² -y² +3x-y

it is necessary to calculate two partial derivatives

 d f(x,y)/d (x)=4x+3

d f(x,y)/d (y)=-2y-1

∀f(x,y) (gradient vector) ------------------- >(4x+3)i+(-2y-1)j

the answer part a) is (4x+3)i+(-2y-1)j

b) The direction of the line passing through the point (1,-2) having the maximum slope

insert x=1 and y=-2 into the gradient vector function

∀f(x,y) (gradient vector) ------------------- >(4*1+3)i+(-2*(-2)-1)j

∀f(x,y)=(7)i+(3)j

the answer part b) is (71+3j)

c) The magnitude of the slope at point (1,-2)

║V║=√(7² +3²)=√58=7.62

the answer part c) is 7.62

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4