If `x=asecthetacostheta,y=bsecthetasinthetaandz=thetatantheta`, then find the value of `(x^(2))/(a^(2))+(y^(2))/(b^(2))-(z^(2))/(c^(2))`

Options

A. a) 1
B. b) `-10`
C. c) 0
D. d) 2

Answer

To solve the problem, we need to find the value of

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} \]

given that

\[ x = a \sec \theta \cos \theta, \quad y = b \sec \theta \sin \theta, \quad z = c \tan \theta. \]

### Step 1: Express \( \frac{x}{a} \), \( \frac{y}{b} \), and \( \frac{z}{c} \)

1. From the definition of \( x \):

\[ \frac{x}{a} = \sec \theta \cos \theta \]

2. From the definition of \( y \):

\[ \frac{y}{b} = \sec \theta \sin \theta \]

3. From the definition of \( z \):

\[ \frac{z}{c} = \tan \theta \]

### Step 2: Substitute these values into the expression

Now we can substitute these values into our expression:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = \left(\sec \theta \cos \theta\right)^2 + \left(\sec \theta \sin \theta\right)^2 - \left(\tan \theta\right)^2 \]

### Step 3: Simplify each term

1. Simplifying \( \left(\sec \theta \cos \theta\right)^2 \):

\[ \left(\sec \theta \cos \theta\right)^2 = \sec^2 \theta \cdot \cos^2 \theta = \frac{1}{\cos^2 \theta} \cdot \cos^2 \theta = 1 \]

2. Simplifying \( \left(\sec \theta \sin \theta\right)^2 \):

\[ \left(\sec \theta \sin \theta\right)^2 = \sec^2 \theta \cdot \sin^2 \theta = \frac{1}{\cos^2 \theta} \cdot \sin^2 \theta \]

3. Simplifying \( \left(\tan \theta\right)^2 \):

\[ \left(\tan \theta\right)^2 = \frac{\sin^2 \theta}{\cos^2 \theta} \]

### Step 4: Combine the terms

Now we can combine the terms:

\[ 1 + \frac{\sin^2 \theta}{\cos^2 \theta} - \frac{\sin^2 \theta}{\cos^2 \theta} \]

Notice that \( \frac{\sin^2 \theta}{\cos^2 \theta} \) cancels out:

\[ 1 + 0 = 1 \]

### Final Result

Thus, the value of

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1. \]