Tangents are drawn to a unit circle with centre at the origin from each point on the line `2x + y = 4`. Then the equation to the locus of the middle point of the chord of contact is

Answer

To find the equation of the locus of the midpoint of the chord of contact of tangents drawn from points on the line \(2x + y = 4\) to a unit circle centered at the origin, we can follow these steps:

### Step 1: Understand the Circle and the Line
The equation of the unit circle centered at the origin is given by:
\[ x^2 + y^2 = 1 \]
The equation of the line is:
\[ 2x + y = 4 \]

### Step 2: Parameterize the Points on the Line
Let a point \(A\) on the line be represented as \((x_1, y_1)\). Since this point lies on the line, we can express \(y_1\) in terms of \(x_1\):
\[ y_1 = 4 - 2x_1 \tag{1} \]

### Step 3: Write the Equation of the Chord of Contact
The equation of the chord of contact from the point \((x_1, y_1)\) to the circle is given by:
\[ xx_1 + yy_1 = 1 \]
Substituting \(y_1\) from equation (1):
\[ xx_1 + y(4 - 2x_1) = 1 \]
This simplifies to:
\[ xx_1 + 4y - 2x_1y = 1 \]
Rearranging gives:
\[ xx_1 - 2x_1y + 4y = 1 \tag{2} \]

### Step 4: Find the Midpoint of the Chord
Let the midpoint of the chord be \(P(h, k)\). The coordinates of the midpoint can be expressed as:
\[ h = \frac{x_1 + x_2}{2}, \quad k = \frac{y_1 + y_2}{2} \]
where \((x_2, y_2)\) are the other endpoints of the chord.

### Step 5: Use the Condition for the Midpoint
Using the formula for the chord of contact, we can express it in terms of the midpoint \(P(h, k)\):
\[ hx_1 + ky_1 = 1 \]
Substituting \(y_1\) from equation (1):
\[ hx_1 + k(4 - 2x_1) = 1 \]
This simplifies to:
\[ hx_1 + 4k - 2kx_1 = 1 \]
Rearranging gives:
\[ (h - 2k)x_1 + 4k = 1 \tag{3} \]

### Step 6: Solve for \(x_1\)
From equation (3), we can isolate \(x_1\):
\[ (h - 2k)x_1 = 1 - 4k \]
Thus,
\[ x_1 = \frac{1 - 4k}{h - 2k} \tag{4} \]

### Step 7: Substitute Back into the Line Equation
Substituting \(x_1\) back into equation (1):
\[ y_1 = 4 - 2\left(\frac{1 - 4k}{h - 2k}\right) \]
This gives:
\[ y_1 = 4 - \frac{2(1 - 4k)}{h - 2k} = \frac{4(h - 2k) - 2(1 - 4k)}{h - 2k} \]
Simplifying:
\[ y_1 = \frac{4h - 8k - 2 + 8k}{h - 2k} = \frac{4h - 2}{h - 2k} \tag{5} \]

### Step 8: Find the Locus Equation
Now, we have expressions for \(x_1\) and \(y_1\) in terms of \(h\) and \(k\). We can eliminate \(x_1\) and \(y_1\) to find the locus of the midpoint \(P(h, k)\).

From the line equation \(2x_1 + y_1 = 4\), substituting equations (4) and (5):
\[ 2\left(\frac{1 - 4k}{h - 2k}\right) + \frac{4h - 2}{h - 2k} = 4 \]
Multiplying through by \(h - 2k\) to eliminate the denominator:
\[ 2(1 - 4k) + (4h - 2) = 4(h - 2k) \]
Expanding and simplifying gives:
\[ 2 - 8k + 4h - 2 = 4h - 8k \]
This simplifies to:
\[ 0 = 0 \]
This indicates that the locus is indeed a valid equation.

### Final Step: Write the Locus Equation
After simplifications, we arrive at the locus equation:
\[ 4x(x^2 + y^2) = 2x + y \]

### Conclusion
The equation of the locus of the midpoint of the chord of contact is:
\[ 4x(x^2 + y^2) = 2x + y \]