Equation of circle passing through `(1,5) and (4,1) and ` touching y-axis is `x ^(2) + y ^(2) - 5x - 6y + 9 + lamda (4x + 3y -19) =0` where `lamda` is equal to

Options

A. `0,-40//9`
B. 0
C. `40//9`
D. `-40//9`

Answer

To find the value of \(\lambda\) for the given equation of the circle that passes through the points \((1,5)\) and \((4,1)\) and touches the y-axis, we can follow these steps:

### Step 1: Substitute \(x = 0\) into the circle equation

The equation of the circle is given as:

\[ x^2 + y^2 - 5x - 6y + 9 + \lambda(4x + 3y - 19) = 0 \]

Since the circle touches the y-axis, we substitute \(x = 0\):

\[ 0^2 + y^2 - 5(0) - 6y + 9 + \lambda(4(0) + 3y - 19) = 0 \]

This simplifies to:

\[ y^2 - 6y + 9 + \lambda(3y - 19) = 0 \]

### Step 2: Rearranging the equation

Rearranging gives:

\[ y^2 + (3\lambda - 6)y + (9 - 19\lambda) = 0 \]

### Step 3: Apply the condition for touching the y-axis

For the circle to touch the y-axis, the discriminant of this quadratic equation must be zero. The discriminant \(D\) is given by:

\[ D = b^2 - 4ac \]

Here, \(a = 1\), \(b = 3\lambda - 6\), and \(c = 9 - 19\lambda\). Thus, we set up the equation:

\[ (3\lambda - 6)^2 - 4(1)(9 - 19\lambda) = 0 \]

### Step 4: Expand and simplify the discriminant

Expanding the discriminant:

\[ (3\lambda - 6)^2 = 9\lambda^2 - 36\lambda + 36 \]
\[ -4(9 - 19\lambda) = -36 + 76\lambda \]

Combining these gives:

\[ 9\lambda^2 - 36\lambda + 36 - 36 + 76\lambda = 0 \]

This simplifies to:

\[ 9\lambda^2 + 40\lambda = 0 \]

### Step 5: Factor the equation

Factoring out \(\lambda\):

\[ \lambda(9\lambda + 40) = 0 \]

### Step 6: Solve for \(\lambda\)

Setting each factor to zero gives us:

1. \(\lambda = 0\)
2. \(9\lambda + 40 = 0 \Rightarrow \lambda = -\frac{40}{9}\)

### Conclusion

The possible values for \(\lambda\) are \(0\) and \(-\frac{40}{9}\).