Locus of the point of intersection of lines `x cosalpha + y sin alpha = a` and `x sin alpha - y cos alpha =b (alpha in R )` is

Options

A. `2(x^(2)+y^(2))= a^(2) + b^(2)`
B. `x^(2) - y^(2) = a^(2 - b^(2)`
C. `x^(2)+y^(2)=a^(2)+b^(2)`
D. `x^(2) - y^(2) = a^(2) +b^(2)`

Answer

To find the locus of the point of intersection of the lines given by the equations \( x \cos \alpha + y \sin \alpha = a \) and \( x \sin \alpha - y \cos \alpha = b \), we can follow these steps:

### Step 1: Set Up the Equations
Let the point of intersection be \( P(h, k) \). This point must satisfy both equations:
1. \( h \cos \alpha + k \sin \alpha = a \) (Equation 1)
2. \( h \sin \alpha - k \cos \alpha = b \) (Equation 2)

### Step 2: Rearranging the Equations
From Equation 1, we can express \( h \):
\[ h = \frac{a - k \sin \alpha}{\cos \alpha} \]
From Equation 2, we can express \( k \):
\[ k = \frac{h \sin \alpha - b}{-\cos \alpha} \]

### Step 3: Square Both Equations
Now, we will square both equations to eliminate \( \alpha \):
1. Squaring Equation 1:
\[ (h \cos \alpha + k \sin \alpha)^2 = a^2 \]
Expanding this gives:
\[ h^2 \cos^2 \alpha + k^2 \sin^2 \alpha + 2hk \cos \alpha \sin \alpha = a^2 \quad \text{(Equation 3)} \]

2. Squaring Equation 2:
\[ (h \sin \alpha - k \cos \alpha)^2 = b^2 \]
Expanding this gives:
\[ h^2 \sin^2 \alpha + k^2 \cos^2 \alpha - 2hk \sin \alpha \cos \alpha = b^2 \quad \text{(Equation 4)} \]

### Step 4: Add the Two Equations
Now, add Equation 3 and Equation 4:
\[ (h^2 \cos^2 \alpha + k^2 \sin^2 \alpha + 2hk \cos \alpha \sin \alpha) + (h^2 \sin^2 \alpha + k^2 \cos^2 \alpha - 2hk \sin \alpha \cos \alpha) = a^2 + b^2 \]
This simplifies to:
\[ h^2 (\cos^2 \alpha + \sin^2 \alpha) + k^2 (\sin^2 \alpha + \cos^2 \alpha) = a^2 + b^2 \]
Using the identity \( \cos^2 \alpha + \sin^2 \alpha = 1 \):
\[ h^2 + k^2 = a^2 + b^2 \]

### Step 5: Replace Variables
Replace \( h \) with \( x \) and \( k \) with \( y \) to find the locus:
\[ x^2 + y^2 = a^2 + b^2 \]

### Final Result
The locus of the point of intersection of the given lines is:
\[ x^2 + y^2 = a^2 + b^2 \]

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