Question 21 of 70advanced🔧 ApplyShort Answer4 marks

A line perpendicular to the line segment joining the points (1,0)(1, 0) and (2,3)(2, 3) divides it in the ratio 1:n1:n. Find the equation of the line.

Correct Answer

The equation of the required line is (n+1)x+3(n+1)y=n+11(n+1)x + 3(n+1)y = n + 11 or equivalently (n+1)x+3(n+1)y−(n+11)=0(n+1)x + 3(n+1)y - (n+11) = 0.

Exercise: EXERCISE 9.2 | Q: 10 | (Page 14)
For More Understanding

Explanation

This question tests the application of section formula and perpendicular slopes. The student needs to: (1) find the coordinates of the point dividing the segment in ratio 1:n using the section formula from the textbook, (2) find the slope of the given line segment, (3) use the negative reciprocal for the perpendicular line's slope, and (4) apply the point-slope form to get the final equation.

Solution Steps

  1. Step 1: Find the point dividing the segment in ratio 1:n Using the section formula, the coordinates of point P dividing the segment joining A(1, 0) and B(2, 3) in ratio 1:n are:

    P=(1×2+n×11+n,1×3+n×01+n)=(n+2n+1,3n+1)P = \left( \frac{1 \times 2 + n \times 1}{1 + n}, \frac{1 \times 3 + n \times 0}{1 + n} \right) = \left( \frac{n + 2}{n + 1}, \frac{3}{n + 1} \right)

  2. Step 2: Find the slope of line segment AB Slope of AB = 3−02−1=31=3\frac{3 - 0}{2 - 1} = \frac{3}{1} = 3

  3. Step 3: Find the slope of the perpendicular line Since the required line is perpendicular to AB, its slope is the negative reciprocal:

    m=−13m = -\frac{1}{3}

  4. Step 4: Write the equation using point-slope form Using point P(n+2n+1,3n+1)P\left( \frac{n + 2}{n + 1}, \frac{3}{n + 1} \right) and slope m=−13m = -\frac{1}{3}:

    y−3n+1=−13(x−n+2n+1)y - \frac{3}{n + 1} = -\frac{1}{3}\left(x - \frac{n + 2}{n + 1}\right)

  5. Step 5: Simplify the equation Multiplying both sides by 3(n+1)3(n+1):

    3(n+1)y−9=−(n+1)x+(n+2)3(n+1)y - 9 = -(n+1)x + (n+2) (n+1)x+3(n+1)y=n+11(n+1)x + 3(n+1)y = n + 11

    Therefore, the equation of the required line is:

    (n+1)x+3(n+1)y−(n+11)=0(n+1)x + 3(n+1)y - (n+11) = 0
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