Respuesta :
Answer:
[tex]A = 4[/tex]
Step-by-step explanation:
The equation of the slope of the tangent line L is obtained by deriving the equation of the hyperbola:
[tex]y = \frac{2}{x}[/tex]
[tex]y'=-2\cdot x^{-2}[/tex]
The numerical value of the slope is:
[tex]y' = -2 \cdot (9)^{-2}\\y' = -\frac{2}{81}[/tex]
The component of the y-axis is:
[tex]y = \frac{2}{9}[/tex]
Now, the tangent line has the following mathematical model:
[tex]y = m \cdot x + b[/tex]
The value of the intercept is found by isolating it within the equation and replacing all known variables:
[tex]b = y - m \cdot x[/tex]
[tex]b = \frac{2}{9}-(-\frac{2}{81} )\cdot (9)\\b = \frac{4}{9}[/tex]
Thus, the tangent line is:
[tex]y = -\frac{2}{81}\cdot x + \frac{4}{9}[/tex]
The vertical distance between a point of the tangent line and the origin is given by the intercept.
[tex]d_{y} = \frac{4}{9}[/tex]
In order to find horizontal distance between a point of the tangent line and the origin, let equalize y to zero and clear x:
[tex]-\frac{2}{81}\cdot x + \frac{4}{9}=0[/tex]
[tex]-\frac{2}{9}\cdot x + 4 = 0[/tex]
[tex]x = 18[/tex]
[tex]d_{x} = 18[/tex]
The area of the triangle is computed by this formula:
[tex]A = \frac{1}{2}\cdot d_{x}\cdot d_{y}[/tex]
[tex]A = \frac{1}{2}\cdot (18)\cdot (\frac{4}{9} )[/tex]
[tex]A = 4[/tex]
The Area of triangle bounded by line and coordinate axes is 4 square unit.
Equation of Tangent line:
Given equation of hyperbola is, [tex]xy=2[/tex]
If x = 9. Then [tex]y=2/x=2/9[/tex]
Slope of tangent line,
[tex]\frac{dy}{dx} =-\frac{2}{x^{2} } =-\frac{2}{81}[/tex]
Equation of line is,
[tex]y-\frac{2}{9} =-\frac{2}{81}(x-9)\\ \\y=-\frac{2}{81}x+\frac{4}{9}[/tex]
y- intercept is, [tex]y=\frac{4}{9}[/tex] and x- intercept, [tex]x=18[/tex]
Area of triangle bounded by line and coordinate axes is,
[tex]A=\frac{1}{2}*18*\frac{4}{9} \\\\Area=4[/tex]
Learn more about the Hyperbola here:
https://brainly.com/question/13955041
