Question 34 of 70intermediate🔧 ApplyShort Answer4 marks

Find the points on the xx-axis, whose distances from the line x3+y4=1\frac{x}{3} + \frac{y}{4} = 1 are 4 units.

Correct Answer

The points on the x-axis are (8, 0) and (-2, 0).

Exercise: EXERCISE 9.3 | Q: 4 | (Page 17)
For More Understanding

Explanation

This question tests the application of the perpendicular distance formula from a point to a line. Points on the x-axis have the form (k, 0). The student needs to convert the line equation to standard form and apply the distance formula, then solve the resulting absolute value equation.

Solution Steps

  1. Step 1: Points on the x-axis have coordinates of the form (k, 0).

  2. Step 2: Convert the line equation to standard form: x/3 + y/4 = 1. Multiplying by 12, we get 4x + 3y = 12, or 4x + 3y - 12 = 0.

  3. Step 3: Apply the perpendicular distance formula. Distance from point (x₁, y₁) to line ax + by + c = 0 is d = |ax₁ + by₁ + c|/a2+b2\sqrt{a² + b²}.

  4. Step 4: For point (k, 0): d = |4k + 3(0) - 12|/16+9\sqrt{16 + 9} = |4k - 12|/5

  5. Step 5: Set distance equal to 4 units: |4k - 12|/5 = 4, which gives |4k - 12| = 20.

  6. Step 6: Solve the absolute value equation: 4k - 12 = 20 or 4k - 12 = -20. This gives k = 8 or k = -2.

  7. Step 7: The required points on the x-axis are (8, 0) and (-2, 0).

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