Question 23 of 70intermediate🔧 ApplyShort Answer3 marks

Find equation of the line passing through the point (2,2)(2, 2) and cutting off intercepts on the axes whose sum is 99.

Correct Answer

The equations of the line are 2x + y - 6 = 0 and x + 2y - 6 = 0.

Exercise: EXERCISE 9.2 | Q: 12 | (Page 14)
For More Understanding

Explanation

This question applies the intercept form of a line from Section 9.3.5 of the textbook. The key insight is that when a line makes intercepts 'a' and 'b' on the axes, its equation is x/a + y/b = 1. The condition that the line passes through (2,2) and has intercepts summing to 9 creates a system of two equations. Solving this system yields a quadratic equation with two valid solutions, meaning two different lines satisfy all given conditions.

Solution Steps

  1. Step 1: Let x-intercept = a and y-intercept = b. Using intercept form: x/a + y/b = 1

  2. Step 2: Given sum of intercepts = 9, so a + b = 9, hence b = 9 - a

  3. Step 3: Since line passes through (2, 2): 2/a + 2/b = 1

  4. Step 4: Substitute b = 9 - a: 2/a + 2/(9-a) = 1

  5. Step 5: Simplify: 18/[a(9-a)] = 1, giving a² - 9a + 18 = 0

  6. Step 6: Solve: (a-3)(a-6) = 0, so a = 3 or a = 6

  7. Step 7: For a=3, b=6: equation is x/3 + y/6 = 1, i.e., 2x + y - 6 = 0. For a=6, b=3: equation is x/6 + y/3 = 1, i.e., x + 2y - 6 = 0

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