Question 25 of 70intermediate🔧 ApplyShort Answer3 marks

The perpendicular from the origin to a line meets it at the point (−2,9)(-2, 9), find the equation of the line.

Correct Answer

The equation of the line is 2x - 9y + 85 = 0.

Exercise: EXERCISE 9.2 | Q: 14 | (Page 14)
For More Understanding

Explanation

This problem uses the concept that if a perpendicular is drawn from the origin to a line, then the line from origin to the foot of perpendicular is perpendicular to the given line. Using the point-slope form from the context, we first find the slope of the perpendicular from origin to (-2, 9), then use the perpendicular slope condition to find the slope of the required line.

Solution Steps

  1. Step 1: Find the slope of the perpendicular from origin (0, 0) to point (-2, 9). Slope = (9 - 0)/(-2 - 0) = -9/2

  2. Step 2: Since this perpendicular line has slope -9/2, the required line is perpendicular to it. Using the condition for perpendicular lines (m₁ × m₂ = -1), slope of required line = 2/9

  3. Step 3: Using the point-slope form y - y₀ = m(x - x₀) with point (-2, 9) and slope 2/9: y - 9 = (2/9)(x + 2)

  4. Step 4: Simplifying: 9(y - 9) = 2(x + 2), which gives 9y - 81 = 2x + 4, or 2x - 9y + 85 = 0

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