Baekhyun B.

asked • 12/28/17

The equations of the diagonals of the square formed by the pair of straight lines 3x^2 + 8xy - 3y^2 =0 and 3x^2 + 8xy - 3y^2 + 2x - 4y - 1= 0 are

Options:
1) x=2y,4x + 2y + 1=0
2) 2x + y =0, 2x = 4y + 1
3) x = 2y, 2x = 4y + 1
4) 2x + y =0, 4x + 2y + 1= 0

Michael J.

The equations in your questions are not straight lines.  Then are ellipticals.
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12/28/17

Sava D.

We need to verify that these are, indeed, strait lines. How can we factor the first equation?
3x2+8xy-3y2=0.
If we fix the variable y, and look at the equation as quadratic equation according to x, we can use the quadratic formula to find the two solutions of the equation.
a = 3,
b = 8y,
c = -3y2
We calculate the discriminant first.
D = b2-4ac
   = (8y)2-4•3•(-3y2)
   = 64y2+36y2
   = 100y2
x1=(-b+√D)/2a
x1=(-8y+√(100y2))/6
x1=(2y)/6
x1=1/3y
x2=-3y.
 
Thus, the first equation is
3x2+8xy-3y2=0,
(x-x1)(x-x2)=0,
(x-1/3y)(x+3y)=0.
The solution set of the equation above is the either all pairs of numbers on the first line (x-1/3y =0) or the second line (x+3y) =0.
 
We can perform the same operation with the second set of lines and obtain
(x-1/3y-1/3)(x+3y+1)=0.
 
Now, you can finish the work finding the answer.
Find the intersection points of the first two and second two lines in each equation above. Use the factored form to find the intersection point.
 
The two points found will be the two endpoints of one of the diagonal. Use them to find the equation of the first diagonal.
 
For the second diagonal, you must mix the lines. One of the systems, for one of the endpoints of the second diagonal will be
x+3y+1=0,
x-1/3y=0.
 
Can you find the second system?
 
After solving these two systems, you will see what equation you will come up with to find the second endpoint.
 
Next, figure out the equation of the second line and you can write the equation of the second line.
You can find the answer you are looking for.
 
Any questions?
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12/28/17

1 Expert Answer

By:

Andy C. answered • 12/28/17

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Options 3 and 4 can be ruled out because the slopes of both lines
are 1/2 and -2 respectively, which means they are parallel.
The diagonals cannot be parallel.
 
In fact, because they are the diagonals of the square, the
base angles are 45 degrees each, which means the
angle of intersection is 90 degrees.
 
In other words, they are perpendicular.
The slopes then must be negative reciprocals of each other.
 
 
Both options (1) and (2) feature perpendicular lines
with slopes of -2 and 1/2. 
 
Once you get the other lines figured out, you can decide
from there which one is correct.
 
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