Let the locus of the mid point of the chords of the circle `x^2+(y-1)^2=` drawn from the origin intersects the line `x+y=1` at P and Q. Then the length of PQ is

Options

A. `1/2`
B. `1/sqrt2`
C. `1/4`
D. `1/sqrt3`

Answer

To solve the problem, we need to find the locus of the midpoints of the chords of the circle defined by the equation \(x^2 + (y - 1)^2 = r^2\) that are drawn from the origin (0, 0) and then determine where this locus intersects the line \(x + y = 1\). Finally, we will calculate the length of the segment \(PQ\) formed by these intersection points.

### Step-by-Step Solution:

1. **Identify the Circle's Equation**:
The given circle is \(x^2 + (y - 1)^2 = r^2\). For simplicity, we can assume \(r^2 = 1\), so the equation becomes:
\[ x^2 + (y - 1)^2 = 1 \]

2. **Find the Equation of the Chord**:
The general equation of a chord of the circle that passes through the origin (0, 0) can be expressed as:
\[ y = mx \]
where \(m\) is the slope of the chord.

3. **Substitute into the Circle's Equation**:
Substitute \(y = mx\) into the circle's equation:
\[ x^2 + (mx - 1)^2 = 1 \]
Expanding this gives:
\[ x^2 + (m^2x^2 - 2mx + 1) = 1 \]
Simplifying, we have:
\[ (1 + m^2)x^2 - 2mx = 0 \]

4. **Factor the Equation**:
Factoring out \(x\):
\[ x((1 + m^2)x - 2m) = 0 \]
This gives \(x = 0\) (the origin) or:
\[ x = \frac{2m}{1 + m^2} \]

5. **Find the Corresponding y-coordinate**:
Substitute \(x\) back into the equation \(y = mx\):
\[ y = m\left(\frac{2m}{1 + m^2}\right) = \frac{2m^2}{1 + m^2} \]

6. **Midpoint of the Chord**:
The midpoint \(M\) of the chord is given by:
\[ M\left(\frac{2m}{1 + m^2}, \frac{2m^2}{1 + m^2}\right) \]

7. **Locus of Midpoints**:
To find the locus, eliminate \(m\):
Let \(x = \frac{2m}{1 + m^2}\) and \(y = \frac{2m^2}{1 + m^2}\). From the first equation, we can express \(m\) in terms of \(x\):
\[ m = \frac{x(1 + m^2)}{2} \]
Substitute this into the second equation and simplify to find the relationship between \(x\) and \(y\).

8. **Intersect with the Line \(x + y = 1\)**:
Substitute \(y = 1 - x\) into the locus equation derived in the previous step. Solve for \(x\) to find the points of intersection \(P\) and \(Q\).

9. **Calculate Length of PQ**:
The length \(PQ\) can be calculated using the distance formula between the two intersection points found in the previous step.