The locus of the point `P(h, k)` for which the line `hx+ky=1` touches the circle `x^(2)+y^(2)=4`, is

Options

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

Explanation

To find the locus of the point \( P(h, k) \) for which the line \( hx + ky = 1 \) touches the circle \( x^2 + y^2 = 4 \), we will follow these steps:

### Step 1: Identify the Circle's Properties
The given circle is \( x^2 + y^2 = 4 \).
- The center of the circle is \( (0, 0) \).
- The radius \( r \) of the circle is \( \sqrt{4} = 2 \).

**Hint:** Remember that the equation of a circle in standard form is \( (x - h)^2 + (y - k)^2 = r^2 \).

### Step 2: Use the Condition of Tangency
For a line to be tangent to a circle, the perpendicular distance from the center of the circle to the line must equal the radius of the circle.

### Step 3: Write the Line in Standard Form
The line given is \( hx + ky = 1 \). We can rewrite it in the standard form \( Ax + By + C = 0 \):
\[ hx + ky - 1 = 0 \]
Here, \( A = h \), \( B = k \), and \( C = -1 \).

### Step 4: Calculate the Perpendicular Distance
The formula for the perpendicular distance \( d \) from a point \( (x_1, y_1) \) to the line \( Ax + By + C = 0 \) is:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Substituting the center of the circle \( (0, 0) \) into the formula:
\[ d = \frac{|h(0) + k(0) - 1|}{\sqrt{h^2 + k^2}} = \frac{|-1|}{\sqrt{h^2 + k^2}} = \frac{1}{\sqrt{h^2 + k^2}} \]

**Hint:** Make sure to use absolute values when calculating distances.

### Step 5: Set the Distance Equal to the Radius
Since the line touches the circle, we set the distance equal to the radius:
\[ \frac{1}{\sqrt{h^2 + k^2}} = 2 \]

### Step 6: Solve for \( h^2 + k^2 \)
Squaring both sides gives:
\[ \frac{1}{h^2 + k^2} = 4 \]
Taking the reciprocal:
\[ h^2 + k^2 = \frac{1}{4} \]

### Conclusion
The locus of the point \( P(h, k) \) is given by the equation of a circle:
\[ h^2 + k^2 = \frac{1}{4} \]
This represents a circle centered at the origin with a radius of \( \frac{1}{2} \).

**Final Answer:** The locus of the point \( P(h, k) \) is the circle \( h^2 + k^2 = \frac{1}{4} \).

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