HI guys.
find the equations of the lines represented by
4x^2 - 5xy - 6y^2 + 11x + 11y - 3 = 0
I am lost. What two lines are they asking for? And is there a general formula/appraoch to simplifying the duel equation with powers to 2?
I tried turning the x^2 and y^2 into squares but then I was left with the 5xy not knowing what to do. Please help.
It is a conic. First use the discriminant theorem to figure out what conic it is
where
A x^2 + B x y + C y^2 + D x + E y + F = 0
if
B^2 -4 A C
negative - ellipse
0 - parabola
positive - hyperbola
here B^2 - 4 A C = 25 - 4*4*(-6)
= 25 + 96 = +121 so it is a hyperbola
the axis of the hyperbola is at an angle to the x axis though, resulting in the xy term with coefficient -5
They may be asking for the directions of x' and y' that make this a decoupled hyperbola with no xy term. I do not know.
Hi! Don't worry, I'm here to help you with finding the equations of the lines represented by the given equation.
To simplify the given equation, we can use a technique called "completing the square." This involves transforming the quadratic equation into a perfect square format, which will help us identify the equations of the lines.
Starting with the given equation: 4x^2 - 5xy - 6y^2 + 11x + 11y - 3 = 0
To complete the square, let's group the terms containing the same variables:
(4x^2 + 11x) - 5xy + (-6y^2 + 11y) = 3
Now, we focus on completing the square for the terms involving x and y separately.
Completing the square for the x-terms:
First, we need to find half of the coefficient of x and square it.
Half of 11 is 11/2, and squaring it gives (11/2)^2 = 121/4.
Adding 121/4 to both sides of the equation gives us:
4x^2 + 11x + 121/4 - 5xy + (-6y^2 + 11y) = 3 + 121/4
Simplifying, we have:
(2x + 11/2)^2 - 5xy + (-6y^2 + 11y) = 3 + 121/4
Next, we complete the square for the y-terms.
Similarly, we find half of the coefficient of y and square it.
Half of 11 is 11/2, and squaring it gives (11/2)^2 = 121/4.
Adding 121/4 to both sides of the equation gives us:
(2x + 11/2)^2 - 5xy + (-6y^2 + 11y) + 121/4 = 3 + 121/4 + 121/4
Simplifying this, we get:
(2x + 11/2)^2 - 5xy - (6y^2 - 11y) + 121/4 = 3 + 242/4
(2x + 11/2)^2 - 5xy - 6(y^2 - (11/6)y) + 121/4 = 3 + 242/4
Now, we can factor the term inside the parentheses:
(2x + 11/2)^2 - 5xy - 6(y - 11/12)^2 + 121/4 - 363/12 + 242/4 = 3 + 242/4
Let's simplify further:
(2x + 11/2)^2 - 5xy - 6(y - 11/12)^2 + 121/4 - 363/12 + 121/4 = 3 + 59/12
We can combine the constant terms on the right side to get:
(2x + 11/2)^2 - 5xy - 6(y - 11/12)^2 + 121/4 - 363/12 + 121/4 = 60/4 + 59/12
Simplifying the constant terms:
(2x + 11/2)^2 - 5xy - 6(y - 11/12)^2 + 121/4 - 363/12 + 121/4 = 180/12 + 59/12
Combining them gives:
(2x + 11/2)^2 - 5xy - 6(y - 11/12)^2 = 239/12
Now, we can express the given equation as the difference of two squares:
(2x + 11/2)^2 - 6(y - 11/12)^2 = 5xy + 239/12
Finally, we can write the equations of the lines represented by the given equation by rearranging the terms:
Line 1: 2x + 11y + 11/2 = 0
Line 2: -2x + 11y + 11/2 = 0
These are the equations of the lines represented by the given equation.
To summarize, completing the square is a useful technique to simplify quadratic equations and identify the equations of the lines. By grouping the x-terms and y-terms separately, we can complete the square for each variable and eventually express the given equation as the difference of two squares. Then, by rearranging the terms, we can obtain the equations of the lines.
I hope this explanation helps! Let me know if you have any further questions.