Question 61 of 70advanced🔧 ApplyShort Answer4 marks

Find the distance of the line 4x+7y+5=04x + 7y + 5 = 0 from the point (1,2)(1, 2) along the line 2x−y=02x - y = 0.

Correct Answer

To find the distance, we determine the point of intersection of the line 4x+7y+5=04x + 7y + 5 = 0 and the line 2x−y=02x - y = 0. Since the point (1,2)(1, 2) satisfies 2x−y=02x - y = 0 (as 2(1)−2=02(1) - 2 = 0), the line passes through the point. We solve the system 4x+7y+5=04x + 7y + 5 = 0 and 2x−y=02x - y = 0.

Substituting y=2xy = 2x into the first equation yields 18x+5=018x + 5 = 0, giving x=−5/18x = -5/18 and y=−5/9y = -5/9. Thus, the intersection point is Q(−5/18,−5/9)Q(-5/18, -5/9).

Using the distance formula, the distance between P(1,2)P(1, 2) and Q(−5/18,−5/9)Q(-5/18, -5/9) is (−23/18)2+(−23/9)2\sqrt{(-23/18)^2 + (-23/9)^2}. Simplifying this gives 5(23/18)2=23518\sqrt{5(23/18)^2} = \frac{23\sqrt{5}}{18} units.

Exercise: Miscellaneous Exercise on Chapter 9 | Q: 14 | (Page 22)
For More Understanding

Explanation

This solution follows the method demonstrated in Example 12 of the provided context. The strategy involves finding the specific point where the line along which the distance is measured intersects the given line. Once the intersection point Q is found by solving the linear equations, the distance is calculated using the distance formula between the original point P and the intersection point Q, as shown in Example 9.

Solution Steps

  1. Step 1: Verify that point P(1, 2) lies on the line 2x - y = 0. Substituting the coordinates: 2(1) - 2 = 0. The point lies on the line.

  2. Step 2: Find the intersection point Q of the lines 4x + 7y + 5 = 0 and 2x - y = 0. Substituting y = 2x into the first equation gives 4x + 14x + 5 = 0, so 18x = -5, or x = -5/18. Then y = -5/9. Point Q is (-5/18, -5/9).

  3. Step 3: Calculate the distance between P(1, 2) and Q(-5/18, -5/9) using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2).

  4. Step 4: Compute differences: x2 - x1 = -5/18 - 1 = -23/18 and y2 - y1 = -5/9 - 2 = -23/9.

  5. Step 5: Calculate distance: d = sqrt((-23/18)^2 + (-23/9)^2) = sqrt(529/324 + 529/81) = sqrt(2645/324) = 23sqrt(5)/18 units.

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