Straight Lines?

Question

We haves some Brain Teasers in chapter known as straight lines
One of them is this,,,any i cant find the answer to this :


Show that the lines given by
3(x)^2 + 8xy - 3(y)^2 = 0 and x-2y-3=0 contain the sides of an ISOCELES RIGHTANGLED TRIANGLE


please help in the proof......

Scythian i dont understand you !!!

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SORRY THE LAST EQUATION IS :

X+2Y=3

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Answer 1

Okay, this proof has been redone in response to the corrected equation for the 3rd line. The proof first shows what are the equations of the 2 lines that form the right angle of the triangle, and the intersection of the 2 lines is at (0,0). Then the 3rd line is drawn, which intersects the other 2 lines. The 3 points of intersection form the triangle. To prove that it is isoceles, the distances from the origin (0,0) to the 2 points of intersection is computed, and shown to be the same. To understand this proof, you have to at least be able to draw out the lines on a x-y plot, or else this will be very hard for you to visualize and understand.

3 x² + 8 xy - 3 y² = 0 factorizes into (3 x - y) (x + 3 y) = 0, which means we have the lines y = - 3 x and y = (1/3) x, which are orthogonal because the slopes are reciprocal. They intersect at (0,0), obviously. So, we look at the line defined by 2y + x = 3, or y = - (1/2) x + (3/2), and find the points of intersection:

y = - 3 x = - (1/2) x + (3/2), or x = - (3/5) and y = (9/5)
y= (1/3) x = - (1/2) x + (3/2), or x = (9/5) and y = (3/5)

By Pythgorean, we find the square of the distances from (0,0)

√((-3/5)² + (9/5)²) = (1/5)√(108)
√((9/5)² + (3/5)²) = (1/5)√(108)

which was obviously true anyway, but that proves that it's isoceles.

Answer 2

3(x)^2 + 8xy - 3(y)^2 = 0
(3x - y)(x + 3y) = 0
x - 2y - 3 = 0
x = 2y + 3, y = (1/2)x - 3/2
(3(2y + 3) - y)(2y + 3 + 3y) = 0
(5y + 9)(5y + 3) = 0
y = -9/5, -3/5
x = -3/5, x = 9/5
midpoint of y = (1/2)x - 3/2 lies at (3/5, -6/5)
┼ bisector is
y = - 2x
(5x)(- 5x) = 0
Intersection of bisector with curve is (0, 0)
The line from (0,0) to (-3/5,-9/5) is perpendicular to the line from (0,0) to (-3/5,9/5). The three points, therefore define an isosceles right triangle.