The method of eliminating `'theta'` from two given equations involving trigonometrical functions of `'theta'`. By using given equations involving `'theta'` and trigonometrical identities, we shall obtain an equation not involving `'theta'`.
On the basis of above information answer the following questions.
If `(x)/(a cos theta)=(y)/(b sin theta)` ...(i)
and `(ax)/(cos theta)- (by )/(sin theta) = a^(2)-b^(2)`, then (x, y) lie on

Options

A. a circle
B. a parabola
C. an ellipse
D. a hyperbola

Answer

To eliminate the variable \( \theta \) from the given equations and find the relationship between \( x \) and \( y \), we will follow these steps:

### Step 1: Write down the given equations
We have the following two equations:
1. \(\frac{x}{a \cos \theta} = \frac{y}{b \sin \theta}\) ...(i)
2. \(\frac{a x}{\cos \theta} - \frac{b y}{\sin \theta} = a^2 - b^2\) ...(ii)

### Step 2: Rearranging the first equation
From equation (i), we can express \( \frac{x}{\cos \theta} \):
\[ \frac{x}{\cos \theta} = \frac{a y}{b \sin \theta} \]

### Step 3: Substitute into the second equation
Now, substitute \( \frac{x}{\cos \theta} \) into equation (ii):
\[ a \left(\frac{a y}{b \sin \theta}\right) - \frac{b y}{\sin \theta} = a^2 - b^2 \]
This simplifies to:
\[ \frac{a^2 y}{b \sin \theta} - \frac{b y}{\sin \theta} = a^2 - b^2 \]

### Step 4: Combine the terms
Combine the terms on the left side:
\[ \frac{(a^2 - b^2) y}{b \sin \theta} = a^2 - b^2 \]

### Step 5: Cancel out \( a^2 - b^2 \)
Assuming \( a^2 - b^2 \neq 0 \), we can cancel \( a^2 - b^2 \) from both sides:
\[ \frac{y}{b \sin \theta} = 1 \]
This gives us:
\[ y = b \sin \theta \]

### Step 6: Substitute back into the first equation
Now substitute \( y \) back into equation (i):
\[ \frac{x}{a \cos \theta} = \frac{b \sin \theta}{b \sin \theta} \]
This simplifies to:
\[ \frac{x}{a \cos \theta} = 1 \]
Thus, we have:
\[ x = a \cos \theta \]

### Step 7: Use the Pythagorean identity
Now we have:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \sin \theta = \frac{y}{b} \) and \( \cos \theta = \frac{x}{a} \):
\[ \left(\frac{y}{b}\right)^2 + \left(\frac{x}{a}\right)^2 = 1 \]

### Step 8: Rearranging the equation
This leads to:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

### Conclusion
The equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) represents an ellipse.