The straight lines `ax+5y=7` and `4x+by=5` intersected in the point (2,-1), the first meets the x-axis in P and Q the second meets the y-axis in Q, then thelenght of PQ is

Options

A. `10sqrt(7)//6`
B. `13//6`
C. `sqrt(149)//6`
D. `sqrt(99)//6`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Determine the values of \( a \) and \( b \)

We know that the lines \( ax + 5y = 7 \) and \( 4x + by = 5 \) intersect at the point \( (2, -1) \). This means that both equations must be satisfied by the point \( (2, -1) \).

1. Substitute \( x = 2 \) and \( y = -1 \) into the first equation:
\[ a(2) + 5(-1) = 7 \implies 2a - 5 = 7 \implies 2a = 12 \implies a = 6 \]

2. Substitute \( x = 2 \) and \( y = -1 \) into the second equation:
\[ 4(2) + b(-1) = 5 \implies 8 - b = 5 \implies b = 3 \]

### Step 2: Write the equations of the lines

Now that we have \( a = 6 \) and \( b = 3 \), we can write the equations of the lines:

1. The first line becomes:
\[ 6x + 5y = 7 \]

2. The second line becomes:
\[ 4x + 3y = 5 \]

### Step 3: Find the intersection points with the axes

1. **Finding point P (x-intercept of the first line)**:
To find the x-intercept, set \( y = 0 \) in the equation \( 6x + 5y = 7 \):
\[ 6x + 5(0) = 7 \implies 6x = 7 \implies x = \frac{7}{6} \]
So, point \( P \) is \( \left(\frac{7}{6}, 0\right) \).

2. **Finding point Q (y-intercept of the second line)**:
To find the y-intercept, set \( x = 0 \) in the equation \( 4x + 3y = 5 \):
\[ 4(0) + 3y = 5 \implies 3y = 5 \implies y = \frac{5}{3} \]
So, point \( Q \) is \( \left(0, \frac{5}{3}\right) \).

### Step 4: Calculate the distance \( PQ \)

The distance \( PQ \) between the points \( P \left(\frac{7}{6}, 0\right) \) and \( Q \left(0, \frac{5}{3}\right) \) can be calculated using the distance formula:
\[ PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ PQ = \sqrt{\left(0 - \frac{7}{6}\right)^2 + \left(\frac{5}{3} - 0\right)^2} \]
\[ = \sqrt{\left(-\frac{7}{6}\right)^2 + \left(\frac{5}{3}\right)^2} \]
\[ = \sqrt{\frac{49}{36} + \frac{25}{9}} \]
To add these fractions, convert \( \frac{25}{9} \) to have a common denominator:
\[ \frac{25}{9} = \frac{100}{36} \]
Now, we can add:
\[ PQ = \sqrt{\frac{49}{36} + \frac{100}{36}} = \sqrt{\frac{149}{36}} = \frac{\sqrt{149}}{6} \]

### Final Answer:
The length of \( PQ \) is \( \frac{\sqrt{149}}{6} \).