A bird is flying along the straight line 
2y−6x=27. In the same plane, an aeroplane starts to fly in a straight line and passes through the point
(4,12). Consider the point where aeroplane starts to fly as origin. If the bird and plane collides then enter the answer as 1 and if not then 0. Note: Bird and aeroplane can be considered to be of negligible size.

To determine if the bird and the aeroplane collide, we need to analyze the equations of their flight paths and see if they intersect.

### Bird's Path:
The equation of the bird's flight path is given by:
\[ 2y - 6x = 27 \]

We can rewrite this equation in slope-intercept form \( y = mx + c \):
\[ 2y = 6x + 27 \]
\[ y = 3x + \frac{27}{2} \]
So, the bird's flight path is:
\[ y = 3x + \frac{27}{2} \]

### Aeroplane's Path:
The aeroplane starts flying from the point (4, 12), which is treated as the origin (0,0) in a translated coordinate system. Thus, the actual point of origin (0,0) is (4,12) in the original coordinate system.

Any straight line passing through the origin (4, 12) can be written in the form:
\[ y - 12 = m(x - 4) \]
where \( m \) is the slope of the line.

### Intersection of the Two Paths:
To find out if the bird and the plane collide, we need to find if there's a common point on both paths. Thus, we set the equations equal to each other:

1. The bird's path: \( y = 3x + \frac{27}{2} \)
2. The plane's path: \( y - 12 = m(x - 4) \)

First, substitute the bird's equation into the plane's equation:
\[ 3x + \frac{27}{2} - 12 = m(x - 4) \]

Simplify:
\[ 3x + \frac{27}{2} - 24 = m(x - 4) \]
\[ 3x - \frac{21}{2} = m(x - 4) \]
\[ 3x - m(x - 4) = \frac{21}{2} \]

Rewrite:
\[ 3x - mx + 4m = \frac{21}{2} \]
\[ x(3 - m) = \frac{21}{2} - 4m \]
\[ x = \frac{\frac{21}{2} - 4m}{3 - m} \]

### Checking for Real Values:
For \( x \) to be real and valid, \( 3 - m \) must be non-zero (i.e., \( m \neq 3 \)). Now, we should verify if this \( x \) produces a corresponding \( y \) value that matches on both lines.

The \( y \) value for the plane:
\[ y = 3x + \frac{27}{2} \]

The \( y \) value for the plane:
\[ y = m(x - 4) + 12 \]

We must check if the derived \( x \) fits both equations. Because the plane's slope \( m \) and bird's slope intersect in finite plane, they will meet if above equation is consistent.

Since there's no contradiction for any real \( m \), both lines intersect (since different slopes will always intersect), thus:

\[ \text{Enter the answer as 1.} \]

So, the bird and the aeroplane do collide.

### Answer:
\[ 1 \]

Would you like any further details or have any questions?

Here are 5 questions you might want to ask:
1. How do you derive the slope-intercept form of a line equation?
2. Can you explain the concept of coordinate system translation?
3. How do you solve simultaneous linear equations?
4. What are the conditions for two lines to intersect?
5. Can you explain how to determine if a point lies on a given line?

Tip: Always verify the intersection by substituting back into both original equations to ensure consistency.

Explore the collision analysis between a bird and an aeroplane in plane geometry. Understand how to determine if two linear paths intersect in a given plane, utilizing slope-intercept form equations. Suitable for high school students preparing for geometry and algebra examinations.

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