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Verified Answer :

So you have to take the derivative first of 9x^2+16y^2=52. 

18x + (32y)(y') = 0 
32y (y') = -18x 

y' = -18x / 32y 

plug in for 0, to find the slope which will get you - 9/16. 
and you know that the tangent of the line formula is y - y1 = m (x -x1), and the problem gave you the points for y1 and x1. 

so the line tangent is y = - 9/16 (x - 2) - 1

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