4.3 Q-29

Question Statement

Determine whether the given points lie on the same side or opposite sides of the specified line.

• Part (a): Points $(0,0)$ and $(-4,7)$ for the line $5x - 7y + 70 = 0$.
• Part (b): Points $(2,3)$ and $(-2,3)$ for the line $3x - 5y + 8 = 0$.

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Background and Explanation

To determine if two points lie on the same or opposite sides of a line:

1. Substitute the coordinates of each point into the line equation to compute the left-hand side (LHS).
2. Compare the signs of the results:
   • If both have the same sign, the points lie on the same side.
   • If the signs are different, the points lie on opposite sides.

Understanding the relationship between points and the equation of a line is key to solving such problems.

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Solution

Part (a): Points $(0,0)$ and $(-4,7)$ for the line $5x - 7y + 70 = 0$

1. Line Equation:

$$5x - 7y + 70 = 0 \tag{1}$$

2. Substitute the first point $(0,0)$ into the LHS:

$$\text{LHS} = 5(0) - 7(0) + 70$$ $$\text{LHS} = 70 \tag{2}$$

The result is positive, so the point $(0,0)$ lies above the line.

3. Substitute the second point $(-4,7)$ into the LHS:

$$\text{LHS} = 5(-4) - 7(7) + 70$$

Simplify step-by-step:

$$\text{LHS} = -20 - 49 + 70$$ $$\text{LHS} = 1 \tag{3}$$

The result is also positive, so the point $(-4,7)$ lies above the line.

4. Conclusion: Both points have the same sign (positive), meaning they are on the same side of the line.

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Part (b): Points $(2,3)$ and $(-2,3)$ for the line $3x - 5y + 8 = 0$

1. Line Equation:

$$3x - 5y + 8 = 0 \tag{1}$$

2. Substitute the first point $(2,3)$ into the LHS:

$$\text{LHS} = 3(2) - 5(3) + 8$$

Simplify step-by-step:

$$\text{LHS} = 6 - 15 + 8$$ $$\text{LHS} = -1 \tag{2}$$

The result is negative, so the point $(2,3)$ lies below the line.

3. Substitute the second point $(-2,3)$ into the LHS:

$$\text{LHS} = 3(-2) - 5(3) + 8$$

Simplify step-by-step:

$$\text{LHS} = -6 - 15 + 8$$ $$\text{LHS} = -13 \tag{3}$$

The result is also negative, so the point $(-2,3)$ lies below the line.

4. Conclusion: Both points have the same sign (negative), meaning they are on the same side of the line.

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Key Formulas or Methods Used

• Line Equation: $ax + by + c = 0$
• Substitution of points into the line equation to calculate the LHS.
• Sign Analysis:
  • Same signs: Points are on the same side.
  • Different signs: Points are on opposite sides.

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Summary of Steps

1. Write the equation of the line.
2. Substitute each point into the LHS of the equation.
3. Simplify to find the LHS values for both points.
4. Compare the signs of the results:
   • Same sign: Points are on the same side.
   • Opposite signs: Points are on opposite sides.