Chords of the curve `4x^(2) + y^(2)- x + 4y = 0` which substand a right angle at the origin pass thorugh a fixed point whose co-ordinates are :

Options

A. `(1/5,-4/5)`
B. `(-1/5,4/5)`
C. `(1/5,4/5)`
D. `(-1/5,-4/5)`

Answer

To solve the problem, we need to find the fixed point through which the chords of the given curve \(4x^2 + y^2 - x + 4y = 0\) that subtend a right angle at the origin pass.

### Step-by-step Solution:

1. **Rearranging the Equation of the Curve**:
The given equation of the curve is:
\[ 4x^2 + y^2 - x + 4y = 0 \]
We can rearrange this equation to express it in a more standard form.

2. **Identifying the General Form of the Chord**:
The equation of a chord of the curve can be expressed as:
\[ ax + by = 1 \]
where \(a\) and \(b\) are constants.

3. **Using the Condition for Right Angles**:
For the chord to subtend a right angle at the origin, we can use the condition that the product of the slopes of the lines forming the chord must be \(-1\). This can be derived from the coefficients of the quadratic terms in the equation of the curve.

4. **Substituting into the Curve's Equation**:
We substitute \(x = \frac{1 - by}{a}\) into the curve's equation:
\[ 4\left(\frac{1 - by}{a}\right)^2 + y^2 - \left(\frac{1 - by}{a}\right) + 4y = 0 \]
This will allow us to express the equation in terms of \(y\) and find a relationship between \(a\) and \(b\).

5. **Simplifying the Equation**:
After substituting and simplifying, we will obtain a quadratic equation in \(y\). The coefficients of this equation will help us set up a condition for the right angle.

6. **Finding the Fixed Point**:
After simplifying, we will find a linear equation relating \(a\) and \(b\). This can be expressed in the form:
\[ a - 4b = 5 \]
Rearranging gives us:
\[ \frac{a}{5} - \frac{4b}{5} = 1 \]
This indicates that the fixed point through which these chords pass can be expressed in terms of \(a\) and \(b\).

7. **Identifying the Fixed Point Coordinates**:
From the equation \(a - 4b = 5\), we can derive the coordinates of the fixed point as:
\[ \left(\frac{1}{5}, -\frac{4}{5}\right) \]

8. **Conclusion**:
The coordinates of the fixed point through which the chords of the curve that subtend a right angle at the origin pass is:
\[ \left(\frac{1}{5}, -\frac{4}{5}\right) \]