4.3 Q-28

Question Statement

Determine whether the given points lie above or below the specified lines.

• Part (a): Point $(5,8)$ and line $2x - 3y + 6 = 0$.
• Part (b): Point $(-7,6)$ and line $4x + 3y - 9 = 0$.

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Background and Explanation

To check whether a point lies above or below a line, substitute the coordinates of the point into the line’s equation. This results in a value on the left-hand side (LHS):

• If the LHS is positive, the point is above the line.
• If the LHS is negative, the point is below the line.
• If the LHS equals zero, the point lies on the line.

The sign of the coefficient of $y$ determines whether the result aligns with the concept of “above” or “below.”

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Solution

Part (a): Point $(5,8)$ and line $2x - 3y + 6 = 0$

1. Line Equation:

$$2x - 3y + 6 = 0 \tag{1}$$

2. Substitute the point $(5,8)$ into the LHS of the equation:

$$\text{LHS} = 2(5) - 3(8) + 6$$

Simplifying step-by-step:

$$\text{LHS} = 10 - 24 + 6$$ $$\text{LHS} = -8$$

3. Since the result is negative and the coefficient of $y$ is negative in the equation, the point lies above the line.

Part (b): Point $(-7,6)$ and line $4x + 3y - 9 = 0$

1. Line Equation:

$$4x + 3y - 9 = 0 \tag{2}$$

2. Substitute the point $(-7,6)$ into the LHS of the equation:

$$\text{LHS} = 4(-7) + 3(6) - 9$$

Simplifying step-by-step:

$$\text{LHS} = -28 + 18 - 9$$ $$\text{LHS} = -19$$

3. Since the result is negative and the coefficient of $y$ is positive in the equation, the point lies below the line.

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Key Formulas or Methods Used

• Line Equation: $ax + by + c = 0$
• Substitution of point $(x, y)$ into the line equation.
• Sign Analysis:
  • LHS > 0: Point lies above the line.
  • LHS < 0: Point lies below the line.
  • LHS = 0: Point lies on the line.

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Summary of Steps

1. Write the equation of the line.
2. Substitute the coordinates of the given point into the LHS of the equation.
3. Simplify the expression to find the result.
4. Compare the sign of the result with the coefficient of $y$:
   • Same signs: Above the line.
   • Opposite signs: Below the line.