If the chord of contact of tangents from a point P(h, k) to the circle `x^(2)+y^(2)=a^(2)` touches the circle `x^(2)+(y-a)^(2)=a^(2)`, then locus of P is

Options

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

Answer

To solve the problem, we need to find the locus of the point \( P(h, k) \) such that the chord of contact of tangents from this point to the circle \( x^2 + y^2 = a^2 \) touches the circle \( x^2 + (y - a)^2 = a^2 \).

### Step-by-step Solution:

1. **Equation of the Circle**:
The first circle is given by the equation:
\[ x^2 + y^2 = a^2 \]
The second circle is given by:
\[ x^2 + (y - a)^2 = a^2 \]
This can be rewritten as:
\[ x^2 + y^2 - 2ay + a^2 = a^2 \implies x^2 + y^2 - 2ay = 0 \]

2. **Chord of Contact**:
The chord of contact from the point \( P(h, k) \) to the first circle is given by:
\[ hx + ky = a^2 \]

3. **Distance from Center to the Chord**:
The center of the second circle is \( (0, a) \) and its radius is \( a \). The distance \( d \) from the center of the second circle to the chord of contact must equal the radius \( a \). The formula for the distance from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Here, the line can be written as:
\[ hx + ky - a^2 = 0 \]
So, \( A = h \), \( B = k \), and \( C = -a^2 \). The distance from the center \( (0, a) \) to the line is:
\[ d = \frac{|h(0) + k(a) - a^2|}{\sqrt{h^2 + k^2}} = \frac{|ka - a^2|}{\sqrt{h^2 + k^2}} \]

4. **Setting the Distance Equal to Radius**:
Since this distance must equal the radius \( a \):
\[ \frac{|ka - a^2|}{\sqrt{h^2 + k^2}} = a \]
Squaring both sides gives:
\[ (ka - a^2)^2 = a^2(h^2 + k^2) \]

5. **Expanding and Rearranging**:
Expanding the left side:
\[ k^2a^2 - 2ka^2 + a^4 = a^2(h^2 + k^2) \]
Rearranging gives:
\[ k^2a^2 - a^2h^2 - a^2k^2 + 2ka^2 - a^4 = 0 \]
Simplifying this results in:
\[ -h^2 + 2k - a = 0 \]
or:
\[ h^2 = 2ak - a^2 \]

6. **Locus of P**:
The equation \( h^2 = 2ak - a^2 \) can be rewritten in terms of \( x \) and \( y \) (where \( h = x \) and \( k = y \)):
\[ x^2 = 2ay - a^2 \]
This represents a parabola.

### Final Locus Equation:
The locus of the point \( P(h, k) \) is given by:
\[ x^2 = 2ay - a^2 \]