`C_1:x^2+y^2=a^2` and `C_2:x^2+y^2=b^2` are two contentric circles. A line through the centre of these circles intersects them at P and respectively. If the lines through P and Q parallel to y-axis and x-axis respectively meet at the point R, then the locus of R is

Options

A. `x^2+y^2=a^2+b^2`
B. `x^2-y^2=a^2-b^2`
C. `x^2/a^2+y^2/b^2=1`
D. `x^2/b^2+y^2/a^2=1`

Answer

To find the locus of the point R, we start with the two concentric circles given by the equations \( C_1: x^2 + y^2 = a^2 \) and \( C_2: x^2 + y^2 = b^2 \).

### Step 1: Identify Points P and Q
Let the line through the center (0,0) intersect circle \( C_1 \) at point \( P \) and circle \( C_2 \) at point \( Q \). We can express the coordinates of these points in terms of a parameter \( \theta \):
- For point \( P \) on circle \( C_1 \):
\[ P = (a \cos \theta, a \sin \theta) \]
- For point \( Q \) on circle \( C_2 \):
\[ Q = (b \cos \theta, b \sin \theta) \]

### Step 2: Determine Coordinates of Point R
The point \( R \) is defined as the intersection of the line through \( P \) parallel to the y-axis and the line through \( Q \) parallel to the x-axis.

- The line through \( P \) parallel to the y-axis has the equation \( x = a \cos \theta \).
- The line through \( Q \) parallel to the x-axis has the equation \( y = b \sin \theta \).

Thus, the coordinates of point \( R \) can be expressed as:
\[ R = (a \cos \theta, b \sin \theta) \]

### Step 3: Find the Locus of Point R
To find the locus of point \( R \), we eliminate the parameter \( \theta \). We can express \( \cos \theta \) and \( \sin \theta \) in terms of \( x \) and \( y \):
- From the coordinates of \( R \):
\[ x = a \cos \theta \implies \cos \theta = \frac{x}{a} \]
\[ y = b \sin \theta \implies \sin \theta = \frac{y}{b} \]

### Step 4: Use the Pythagorean Identity
We know from trigonometry that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting the expressions for \( \sin \theta \) and \( \cos \theta \):
\[ \left(\frac{y}{b}\right)^2 + \left(\frac{x}{a}\right)^2 = 1 \]

### Step 5: Rearranging the Equation
Multiplying through by \( a^2b^2 \) gives:
\[ y^2 a^2 + x^2 b^2 = a^2 b^2 \]
This represents the equation of an ellipse.

### Final Result
The locus of the point \( R \) is given by the equation:
\[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \]