Straight Lines and Pair of Straight Lines

Step 1: Write the coordinates
Let P be any point (x,y).A is (1,0).B is (0,−2).C is (2,−1).
Step 2: Find area of triangle PAB
The area of triangle PAB is given by:
Area of △PAB=‌

1

2

|x1(y2−y3)+x2(y3−y1)+x3(y1−y2)| For P(x,y),A(1,0),B(0,−2) :
‌ Area ‌=‌

1

2

|x(0+2)+1(−2−y)+0(y−0)|=‌

1

2

|2x−2−y|
Step 3: Find area of triangle PAC
Use the same formula for P(x,y),A(1,0),C(2,−1) :
‌ Area ‌=‌

1

2

|x(0+1)+1(−1−y)+2(y−0)|=‌

1

2

|x+y−1|
Step 4: Set the areas equal
We want these two areas to be the same: ‌

1

2

|2x−2−y|=‌

1

2

|x+y−1| Take both sides times 2:|2x−2−y|=|x+y−1|
Step 5: Remove the modulus (absolute value)
Square both sides to get rid of absolute values: (2x−2−y)2=(x+y−1)2
Step 6: Expand both sides
Expand the left side: (2x−2−y)2=(2x−y−2)2=4x2+y2+4+4xy−8x−4y
Expand the right side: (x+y−1)2=x2+y2+1+2xy−2x−2y
Step 7: Move all terms to one side
Subtract right side from left side:
‌(4x2+y2+4+4xy−8x−4y)−(x2+y2+1+2xy−2x−2y)=0‌ Simplify: ‌
‌4x2−x2+4xy−2xy−8x+2x+(−4y+2y)+4−1=03x2+2xy−6x−2y+3=0
Step 8: Divide by 3
To make it simpler, divide all terms by 3: x2−2xy−2x+2y+1=0