Question 39 of 70intermediate🔧 ApplyShort Answer4 marks

The line through the points (h,3)(h, 3) and (4,1)(4, 1) intersects the line 7x−9y−19=07x - 9y - 19 = 0. at right angle. Find the value of hh.

Correct Answer

h = 22/9

Exercise: EXERCISE 9.3 | Q: 9 | (Page 17)
For More Understanding

Explanation

This question tests the concept of perpendicular lines and their slopes. The key insight is that when two lines are perpendicular, the product of their slopes equals -1. The student must first find the slope of the line through the given points in terms of h, then find the slope of the given line, and finally apply the perpendicular condition to solve for h.

Solution Steps

  1. Step 1: Find the slope of the line through points (h, 3) and (4, 1). The slope m₁ = (1-3)/(4-h) = -2/(4-h).

  2. Step 2: Find the slope of the line 7x - 9y - 19 = 0. Rewriting: 9y = 7x - 19, so y = (7/9)x - 19/9. The slope m₂ = 7/9.

  3. Step 3: Apply the perpendicular condition. Since the lines are perpendicular, m₁ × m₂ = -1.

  4. Step 4: Substitute and solve: (-2/(4-h)) × (7/9) = -1. This gives -14/(36-9h) = -1.

  5. Step 5: Simplify: -14 = -(36-9h), so -14 = -36 + 9h, which gives 22 = 9h.

  6. Step 6: Therefore, h = 22/9.

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