If `theta` is eliminated from the equations `a sec theta-x tan theta=y" and "b sec theta+y tan theta=x` (a and b are constants), then :

Options

A. the director circle of the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1`
B. auxiliary circle of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1`
C. director circle of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1`
D. director circle of the circle `x^2+y^2=(a^2+b^2)/(2)`

Answer

To eliminate \( \theta \) from the given equations \( a \sec \theta - x \tan \theta = y \) and \( b \sec \theta + y \tan \theta = x \), we will follow these steps:

### Step 1: Rewrite the equations using trigonometric identities
The first equation is:
\[ a \sec \theta - x \tan \theta = y \]
Using the identities \( \sec \theta = \frac{1}{\cos \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we can rewrite this as:
\[ \frac{a}{\cos \theta} - x \frac{\sin \theta}{\cos \theta} = y \]
Multiplying through by \( \cos \theta \):
\[ a - x \sin \theta = y \cos \theta \]
Rearranging gives:
\[ x \sin \theta + y \cos \theta = a \tag{1} \]

### Step 2: Rewrite the second equation
The second equation is:
\[ b \sec \theta + y \tan \theta = x \]
Again using the identities:
\[ \frac{b}{\cos \theta} + y \frac{\sin \theta}{\cos \theta} = x \]
Multiplying through by \( \cos \theta \):
\[ b + y \sin \theta = x \cos \theta \]
Rearranging gives:
\[ x \cos \theta - y \sin \theta = b \tag{2} \]

### Step 3: Square and add the two equations
Now we will square both equations and add them:
\[ (x \sin \theta + y \cos \theta)^2 + (x \cos \theta - y \sin \theta)^2 = a^2 + b^2 \]
Expanding both sides:
\[ (x^2 \sin^2 \theta + 2xy \sin \theta \cos \theta + y^2 \cos^2 \theta) + (x^2 \cos^2 \theta - 2xy \sin \theta \cos \theta + y^2 \sin^2 \theta) = a^2 + b^2 \]
Combining like terms:
\[ x^2 (\sin^2 \theta + \cos^2 \theta) + y^2 (\sin^2 \theta + \cos^2 \theta) = a^2 + b^2 \]
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ x^2 + y^2 = a^2 + b^2 \]

### Step 4: Conclusion
The final equation \( x^2 + y^2 = a^2 + b^2 \) represents the equation of the director circle of an ellipse.