an ellipse has the equation

That isn't the equation of an ellipse...it's the equation of a hyperbola....

Weird, I couldn't type "x is smaller than n and larger than m" and "x is smaller than 100 and larger than 0" here, now that you say that you can't type "k is smaller than 13 and larger than -13" I think it's related.

I think you need to put spaces between the signs, like this: -13 < k < 13

x^2 / 9  +  y^2 /25  = 1    can be transformed to  25x^2 + 9y^2  = 225    (1)

y  = 4x + k    (2)

The slope of a tangent line at any point on (1)  can be found as

50x + 18y y'  = 0

y'  =  -50x / [ 18y]   =  -25x / [ 9y]

And we are looking for  where the slope of a tangent line  = 4

So

-25x / 9y  = 4

-25x = 36y

y = (-25)/(36) x       sub this into  (1) for y

25x^2  + 9 (-25/36 x)^2 =  225

25x^2  + 9 (625/1296)x^2  = 225

4225/144 x^2  = 225 

x^2  =225 * 144 / 4225     take both roots

x = 15 * 12 / 65  = 180/65  =36/13

Or

x = -36/13

Subbing either value into  (1) to find  y we have

25 (36/13)^2  + 9y^2  = 225

32400 / 169  + 9y^2  = 225

32400 / 169 + 9y^2  = 38025/169

y^2  = [38025 - 32400 ] / [ 9 * 169]

y^2 = [5625] / [ 9 * 169]      take both roots

y = 75 / [ 3 * 13 ] =  75 / 39  = 25 / 13

OR

y = -25/13

So....the slope of the tangent line to the ellipse = 4  at   (-36/13 , 25/13)  and (36/13. -25/13)

Writing an equation of one tangent line using the first point we have

y  = 4 ( x + 36/13)  + 25/13

y = 4x + 144/13 + 25/13

y = 4x   + 169/13

y = 4x + 13

And writing the equation of the other tangent line we have that

y = 4 (x - 36/13) - 25/13

y = 4x - 144/13  - 25/13

y = 4x -169/13

y = 4x - 13

Note the graph here :  https://www.desmos.com/calculator/syelncmges

When k  = 0 .....the graph intersects the ellipse at two points

However when k < -13  ...the tangent line is shifted to the right of the ellipse

And when k > 13....the tangent line is shifted to the left of the ellipse