Xsin(theta)-Ysin(theta)=sqrtX^2+Y^2 and cos^2(theta)/a^2+sin^2(theta)/b^2=1/x^2+Y^2 then find the relation

To find the relationship between X and Y, we will solve the given equations step-by-step.

Equation 1: Xsin(theta) - Ysin(theta) = sqrt(X^2 + Y^2)

Step 1: Factor out sin(theta) on the left side:
sin(theta)(X - Y) = sqrt(X^2 + Y^2)

Step 2: Square both sides:
[sin(theta)(X - Y)]^2 = (sqrt(X^2 + Y^2))^2

Step 3: Simplify:
sin^2(theta)(X - Y)^2 = X^2 + Y^2

Step 4: Distribute sin^2(theta) on the left side:
sin^2(theta)(X^2 - 2XY + Y^2) = X^2 + Y^2

Step 5: Expand and rearrange:
X^2*sin^2(theta) - 2XY*sin^2(theta) + Y^2*sin^2(theta) = X^2 + Y^2

Step 6: Group like terms:
(X^2 - X^2*sin^2(theta)) + (Y^2 - Y^2*sin^2(theta)) = 2XY*sin^2(theta)

Step 7: Factor out common terms:
X^2(1 - sin^2(theta)) + Y^2(1 - sin^2(theta)) = 2XY*sin^2(theta)

Step 8: Remembering the trigonometric identity sin^2(theta) + cos^2(theta) = 1:
X^2*cos^2(theta) + Y^2*cos^2(theta) = 2XY*sin^2(theta)

Equation 2: cos^2(theta)/a^2 + sin^2(theta)/b^2 = 1/(X^2 + Y^2)

Step 9: Multiply both sides by (X^2 + Y^2):
cos^2(theta)*(X^2 + Y^2)/a^2 + sin^2(theta)*(X^2 + Y^2)/b^2 = 1

Step 10: Expand terms:
X^2*cos^2(theta)/a^2 + Y^2*cos^2(theta)/a^2 + X^2*sin^2(theta)/b^2 + Y^2*sin^2(theta)/b^2 = 1

Step 11: Substitute X^2*cos^2(theta) + Y^2*cos^2(theta) = 2XY*sin^2(theta) from equation 1:
(2XY*sin^2(theta))/a^2 + X^2*sin^2(theta)/b^2 + Y^2*sin^2(theta)/b^2 = 1

Step 12: Combine like terms:
(2XY*sin^2(theta))/a^2 + (X^2 + Y^2)*sin^2(theta)/b^2 = 1

Step 13: Multiply both sides by a^2b^2:
2XY*sin^2(theta)*b^2 + (X^2 + Y^2)*sin^2(theta)*a^2 = a^2b^2

Step 14: Factor out sin^2(theta):
sin^2(theta)*(2XY*b^2 + (X^2 + Y^2)*a^2) = a^2b^2

Step 15: Divide both sides by (2XY*b^2 + (X^2 + Y^2)*a^2):
sin^2(theta) = a^2b^2 / (2XY*b^2 + (X^2 + Y^2)*a^2)

Therefore, the relationship between X and Y is given by sin^2(theta) = a^2b^2 / (2XY*b^2 + (X^2 + Y^2)*a^2).

To find the relation between X and Y, we need to manipulate the given equations and simplify them as much as possible. Let's go step by step:

Equation 1: Xsin(theta) - Ysin(theta) = sqrt(X^2 + Y^2)
Equation 2: cos^2(theta)/a^2 + sin^2(theta)/b^2 = 1/x^2 + Y^2

To simplify the equations, we can use trigonometric identities:

From Equation 1:
Xsin(theta) - Ysin(theta) = sqrt(X^2 + Y^2)
sin(theta)(X - Y) = sqrt(X^2 + Y^2)

From Equation 2:
cos^2(theta)/a^2 + sin^2(theta)/b^2 = 1/x^2 + Y^2
(cos^2(theta))(1/a^2) + (sin^2(theta))(1/b^2) = 1/x^2 + Y^2

Now, we know that:
cos^2(theta) + sin^2(theta) = 1

Substituting this into Equation 2:

(1/a^2) + (sin^2(theta))(1/b^2) = 1/x^2 + Y^2

Multiplying both sides by a^2, we get:

1 + (sin^2(theta))(a^2/b^2) = a^2/x^2 + a^2Y^2

Now, let's substitute the value of sin^2(theta) from Equation 1:

1 + (sqrt(X^2 + Y^2))^2(a^2/b^2) = a^2/x^2 + a^2Y^2

Simplifying this equation further:

1 + (X^2 + Y^2)(a^2/b^2) = a^2/x^2 + a^2Y^2

Multiplying both sides by b^2:

b^2 + (X^2 + Y^2)a^2 = a^2b^2/x^2 + a^2b^2Y^2

Now, we can see that the relation between X and Y is given by:

b^2 + (X^2 + Y^2)a^2 = a^2b^2/x^2 + a^2b^2Y^2

This is the final relation between X and Y.