To find the points on the graph of y = 1/x where the graph is parallel to the line 4x + 9y = 3, we need to determine the slope of the given line.
The equation of a line in the form Ax + By = C can be written in slope-intercept form, y = -A/Bx + C/B. Comparing this with 4x + 9y = 3, we can see that its slope is -4/9.
Since we want to find points on the graph of y = 1/x that have the same slope, we can differentiate this equation implicitly. Differentiating y with respect to x, we get:
dy/dx = -1/x^2
To find the slope of the graph of y = 1/x at any given point, we substitute the x-coordinate of the point into dy/dx. Therefore, we need to find where dy/dx equals -4/9:
-1/x^2 = -4/9
Cross-multiplying, we get:
-9 = -4x^2
Dividing both sides by -4, we have:
9/4 = x^2
Taking the square root of both sides, we get:
x = ±sqrt(9/4) = ±3/2
So, the x-coordinates of the points where the graph of y = 1/x is parallel to the line 4x + 9y = 3 are x = -3/2 and x = 3/2.
Substituting these x-coordinates back into the equation y = 1/x, we can find the corresponding y-coordinates:
When x = -3/2, y = 1/(-3/2) = -2/3
When x = 3/2, y = 1/(3/2) = 2/3
Therefore, the points on the graph of y = 1/x that are parallel to the line 4x + 9y = 3 are (-3/2, -2/3) and (3/2, 2/3).