If the tangents and normal to an ellipse `9x^(2) + 16y^(2) = 144` at a point intercept `a_(1), a_(2)` on x-axis and `b_(1), b_(2)` on y-axis. Then the value of `a_(1)a_(2) + b_(1)b_(2) + 1000` is _________ .

Explanation

To solve the problem, we need to find the value of \( a_1 a_2 + b_1 b_2 + 1000 \) given the ellipse \( 9x^2 + 16y^2 = 144 \).

### Step 1: Rewrite the equation of the ellipse
First, we rewrite the equation of the ellipse in standard form:
\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]
This shows that the semi-major axis \( a = 4 \) (along the x-axis) and the semi-minor axis \( b = 3 \) (along the y-axis).
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