9514 1404 393
Answer:
- x = 3.625; x = 4.375
- y = 0; y = -3
Step-by-step explanation:
We can rewrite the equation to standard form to see the center and semi-axis lengths.
16x^2 -128x +y^2 +3y = -256
16(x^2 -8x +16) +(y^2 +3y +2.25) = -256 +256 +2.25
16(x -4)^2 +(y +1.5)^2 = 9/4 . . . . . write as squares
(x -4)^2/(9/64) +(y +1.5)^2/(9/4) = 1 . . . . divide by 9/4
((x -4)/0.375)^2 +((y +1.5)/1.5)^2 = 1 . . . . put in useful form
In this form, we have ...
((x -h)/a)^2 +((y -k)/b)^2 = 1
where (h, k) is the center, 2a is the length of the axis in the x-direction, and 2b is the length of the axis in the y-direction. The required tangents are ...
x = h±a
y = k±b
For the given ellipse, the tangent lines are ...
- x = 4 -0.375 = 3.625, x = 4.375
- y = -1.5 -1.5 = -3, y = -1.5 +1.5 = 0
