Question 44 of 70intermediate🔧 ApplyShort Answer4 marks

The perpendicular from the origin to the line y=mx+cy = mx + c meets it at the point (−1,2)(-1, 2). Find the values of mm and cc.

Correct Answer

The perpendicular line passes through the origin (0,0)(0,0) and the point (−1,2)(-1, 2). Using the method shown in Example 3 of the context, the slope of the perpendicular line is calculated as 2−0−1−0=−2\frac{2-0}{-1-0} = -2.

Since the line y=mx+cy = mx + c is perpendicular to this line, the product of their slopes must be −1-1. Using the condition m1m2=−1m_1 m_2 = -1 from the context, we write m×(−2)=−1m \times (-2) = -1. Solving this, we get m=12m = \frac{1}{2}.

The line y=mx+cy = mx + c passes through the point (−1,2)(-1, 2). Substituting x=−1x = -1, y=2y = 2, and m=12m = \frac{1}{2} into the equation, we get 2=12(−1)+c2 = \frac{1}{2}(-1) + c. Solving for cc, we find c=52c = \frac{5}{2}.

Exercise: EXERCISE 9.3 | Q: 14 | (Page 18)
For More Understanding

Explanation

The solution relies on two key concepts found in the provided context. First, the method for calculating the slope of a line passing through two points is demonstrated in Example 3. Second, the condition for two lines being perpendicular (m1m2=−1m_1 m_2 = -1) is explicitly stated in Example 3. By combining these, we find the slope of the original line. Finally, the definition of the line equation y=mx+cy = mx + c from Case I is used to find the y-intercept cc by substituting the known point coordinates.

Solution Steps

  1. Step 1: Calculate the slope of the perpendicular line passing through (0,0)(0,0) and (−1,2)(-1, 2) as 2−0−1−0=−2\frac{2-0}{-1-0} = -2.

  2. Step 2: Use the perpendicularity condition m×(−2)=−1m \times (-2) = -1 to find m=12m = \frac{1}{2}.

  3. Step 3: Substitute the point (−1,2)(-1, 2) and slope mm into the equation y=mx+cy = mx + c to get 2=12(−1)+c2 = \frac{1}{2}(-1) + c.

  4. Step 4: Solve for cc to get c=52c = \frac{5}{2}.

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