The second you hear tangent line, think derivative. Since this has a in it, we're going to differentiate implicity

Moving stuff around,

So the slopes of our lines must have the form above to be tangent

Now we have 1 point, so let's find the toher point

Points on the elipse are represented by or

And slope=

Sooooo,

The derivative equation must equal the slope between the 2 points and lets use our formulas to plug in for y

And don't forget that the y values can be represented 2 ways (+ and -)

Solving both of those will get you your x values, and then to get your y's remember to keep to + and -'s straight

Then the equation of the tangent lines will be

Of course, you already said 1 tangent line will be horizontal, and you know the point it goes through, so you can get that equation almost without thinking