Please help me to find the separate equations of the line represented by following equations:
a) x^2+2(cosec a)xy-y^2=0
b) x^2+2xy. tan a-y^2=0
These two equations do not represent a (single) line. They are two different curves. For certain values of a, they overlap.
a) x^2-y^2 = 0
represents the lines y = ±x
Ax^2+Bxy+Cy^2=0 represents the same lines rotated through an angle θ such that
cot2θ = C-A/2B
so, we have cot2θ = 1/2 sin(a)
similarly for (b)
To find the separate equations of the lines represented by the given equations, we need to express the equations in the standard form of a line, which is:
ax + by + c = 0
where a, b, and c are constants.
Let's solve each equation separately:
a) x^2 + 2(cosec a)xy - y^2 = 0
Step 1: Rearrange the equation by gathering the terms involving x and y on one side:
x^2 + 2(cosec a)xy - y^2 = 0
=> x^2 - y^2 + 2(cosec a)xy = 0
Step 2: Factorize the left side:
(x - y)(x + y) + 2(cosec a)xy = 0
Step 3: Divide both sides by 2(cosec a) to isolate xy:
[(x - y)(x + y)] / 2(cosec a) + xy = 0
Step 4: Multiply both sides by 2(cosec a) to eliminate the denominator:
(x - y)(x + y) + 2(cosec a)xy = 0
Step 5: Distribute to expand the equation:
x^2 - y^2 + x^2y - y^2x = 0
Step 6: Rearrange the equation to separate the terms involving x and y:
x^2 - y^2 = y^2x - x^2y
Step 7: Divide both sides by xy:
(x^2 - y^2) / xy = (y^2x - x^2y) / xy
Step 8: Simplify each side:
(x / y) - (y / x) = y - x
The separate equations of the lines represented by the equation x^2 + 2(cosec a)xy - y^2 = 0 are:
x / y - y / x = y - x
b) x^2 + 2xy tan a - y^2 = 0
Step 1: Rearrange the equation by gathering the terms involving x and y on one side:
x^2 + 2xy tan a - y^2 = 0
Step 2: Factorize the left side:
(x - y)(x + y) + 2xy tan a = 0
Step 3: Divide both sides by 2tan a to isolate xy:
[(x - y)(x + y)] / 2tan a + xy = 0
Step 4: Multiply both sides by 2tan a to eliminate the denominator:
(x - y)(x + y) + 2xy tan a = 0
Step 5: Distribute to expand the equation:
x^2 - y^2 + x^2 tan a - y^2 tan a = 0
Step 6: Rearrange the equation to separate the terms involving x and y:
x^2 - y^2 = y^2 tan a - x^2 tan a
Step 7: Divide both sides by tan a:
(x^2 - y^2) / tan a = (y^2 tan a - x^2 tan a) / tan a
Step 8: Simplify each side:
x^2 / tan a - y^2 / tan a = y^2 - x^2
The separate equations of the lines represented by the equation x^2 + 2xy tan a - y^2 = 0 are:
x^2 / tan a - y^2 / tan a = y^2 - x^2