Find the coordinates of the points of intersection of the straight line x-y=5 and the hyperbola xy=9
y = 9/x
x - 9/x = 5
x^2 -5 x - 9 = 0
x = [ 5 +/- sqrt (25 + 36) ] / 2 etc
Well, it looks like we've got ourselves an "intersection" situation! Let's see if we can find those points of intersection by solving this little math mystery.
First, let's look at the straight line equation x - y = 5. To make our life a bit easier, let's rewrite it in terms of one variable:
x = 5 + y
Now, let's plug this into the equation of the hyperbola xy = 9, like a matchmaker trying to find some common ground. We'll have:
(5 + y) * y = 9
Expanding and rearranging this equation will lead us closer to discovering the coordinates of those intersection points. After some simplification, we get:
y^2 + 5y - 9 = 0
Now, we have a good old quadratic equation! Using our math toolbox, we can solve for the value(s) of y. After applying the quadratic formula, we find:
y = (-5 ± √(5^2 - 4 * 1 * -9)) / (2 * 1)
Simplifying that mess, we have:
y = (-5 ± √(25 + 36)) / 2
y = (-5 ± √61) / 2
So, we have two possible values for y - let's say y₁ and y₂. Now, we just need to find their corresponding x-values. For that, we plug these values into the equation x = 5 + y:
For y = (-5 + √61) / 2:
x = 5 + [(-5 + √61) / 2]
For y = (-5 - √61) / 2:
x = 5 + [(-5 - √61) / 2]
And voilà! These two pairs of values (x, y) represent the coordinates of the points of intersection between the given straight line and hyperbola. I hope my little math circus managed to bring a smile to your face!
To find the coordinates of the points of intersection of the straight line x - y = 5 and the hyperbola xy = 9, we need to solve these two equations simultaneously.
Step 1: Solve the first equation for y in terms of x:
x - y = 5
y = x - 5
Step 2: Substitute this value of y into the second equation:
x(x - 5) = 9
x^2 - 5x - 9 = 0
Step 3: Solve this quadratic equation. We can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -5, and c = -9. Substituting these values into the formula, we find:
x = (5 ± √((-5)^2 - 4(1)(-9))) / 2(1)
x = (5 ± √(25 + 36)) / 2
x = (5 ± √61) / 2
So, we have two potential x-values for the points of intersection: (5 + √61) / 2 and (5 - √61) / 2.
Step 4: Substitute these x-values back into the equation y = x - 5 to find the corresponding y-values.
For x = (5 + √61) / 2:
y = (5 + √61) / 2 - 5
y = (5 + √61 - 10) / 2
y = (5 - 10 + √61) / 2
y = (-5 + √61) / 2
For x = (5 - √61) / 2:
y = (5 - √61) / 2 - 5
y = (5 - √61 - 10) / 2
y = (5 - 10 - √61) / 2
y = (-5 - √61) / 2
So, the coordinates of the points of intersection are:
( (5 + √61) / 2, (-5 + √61) / 2 )
( (5 - √61) / 2, (-5 - √61) / 2 )
To find the coordinates of the points of intersection between the straight line and the hyperbola, we need to solve the system of equations formed by the line and the hyperbola.
Given:
Straight line: x - y = 5 ------(1)
Hyperbola: xy = 9 ------(2)
We can begin by solving equation (1) for x in terms of y:
x = y + 5 ------(3)
Substituting equation (3) into equation (2), we get:
(y + 5) * y = 9
This can be rewritten as a quadratic equation:
y^2 + 5y - 9 = 0
To solve this quadratic equation, we can use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
Applying the quadratic formula to our equation, we have:
y = (-5 ± √(5^2 - 4 * 1 * (-9))) / (2 * 1)
y = (-5 ± √(25 + 36)) / 2
y = (-5 ± √61) / 2
So we have two possible values for y: y1 = (-5 + √61) / 2 and y2 = (-5 - √61) / 2.
Now, we can substitute these values of y back into equation (3) to find the corresponding x-values.
For y1 = (-5 + √61) / 2:
x1 = y1 + 5 = (-5 + √61) / 2 + 5
For y2 = (-5 - √61) / 2:
x2 = y2 + 5 = (-5 - √61) / 2 + 5
Thus, the coordinates of the points of intersection are:
(x1, y1) = [(-5 + √61) / 2 + 5, (-5 + √61) / 2]
(x2, y2) = [(-5 - √61) / 2 + 5, (-5 - √61) / 2]