Given equation of straight lines is 6y2 - xy - x2 + 30y + 36 = 0.

To find the angle between two straight lines, we need to find the slope of both the lines and then use the formula:

tan θ = |(m2 - m1) / (1 + m1m2)|, where m1 and m2 are the slopes of the two lines.

Steps:

1. Convert the given equation into the standard form of the equation of a straight line, y = mx + c.

2. Find the slope of the first line by equating the coefficient of x to the coefficient of y in the standard form equation.

3. Find the slope of the second line in the same way.

4. Substituting the values of m1 and m2 in the formula, we can find the angle between the two lines.

Let's solve the problem step by step.

Step 1: Convert the given equation into the standard form of the equation of a straight line, y = mx + c.

To do this, we need to group the terms containing y and x together.

6y2 + 30y - x(y + 6) - x2 + 36 = 0

Rearranging the terms, we get:

6y2 + 30y - x(y + 6) - x2 = -36

6y2 + 30y + 36 = x2 + xy + 6x

Dividing both sides by 6, we get:

y2 + 5y + 6 = (1/6)x2 + (1/6)xy + x

y2 + 5y + 6 = (1/6)x(x + y + 6)

Now we can write this equation in the standard form y = mx + c.

y2 + 5y + 6 = (1/6)x(x + y + 6)

6y2 + 30y + 36 = x2 + xy + 6x

x2 + xy + 6x - 6y2 - 30y - 36 = 0

(1/6)x(x + y + 6) - y - 1 = 0

(1/6)x(x + y + 6) = y + 1

y = (1/6)x(x + y + 6) - 1

y = (1/6)x2 + (1/6)xy + x/6 - 1

Comparing this with y = mx + c, we get:

m = (1/6)x + (1/6)y + (1/6)

c = -1

So, the equation of the first line is y = (1/6)x + (1/6)y + (1/6)x - 1.

Step 2: Find the slope of the first line by equating the coefficient of x to the coefficient of y in the standard form equation.

Slope of the first line m1 = (1/6)

Step 3: Find the slope of the second line in the same way.

We need to find the two values of x that satisfy the given equation, and then find the corresponding values of y.

6y2 - xy - x2 + 30y + 36 = 0

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