Question 29 of 70advanced🔧 ApplyShort Answer4 marks

Point R(h,k)R(h, k) divides a line segment between the axes in the ratio 1:21:2. Find equation of the line.

Correct Answer

The equation of the line is 2xh+yk=3\frac{2x}{h} + \frac{y}{k} = 3 or equivalently 2kx+hy=3hk2kx + hy = 3hk.

Exercise: EXERCISE 9.2 | Q: 18 | (Page 14)
For More Understanding

Explanation

This question requires finding the equation of a line when a point divides the segment between the axes in a given ratio. Similar to Question 17 in the context where the midpoint gives equation x/a + y/b = 2, here we use the section formula with ratio 1:2. The key is to find the intercepts on the axes using the section formula, then substitute into the intercept form of a line.

Solution Steps

  1. Step 1: Let the line intersect the x-axis at point A(a, 0) and the y-axis at point B(0, b).

  2. Step 2: Point R(h, k) divides the line segment AB in the ratio 1:2. Using the section formula, the coordinates of R are: h=1⋅0+2⋅a1+2=2a3h = \frac{1 \cdot 0 + 2 \cdot a}{1+2} = \frac{2a}{3} and k=1⋅b+2⋅01+2=b3k = \frac{1 \cdot b + 2 \cdot 0}{1+2} = \frac{b}{3}

  3. Step 3: Solving for intercepts a and b: a=3h2a = \frac{3h}{2} and b=3kb = 3k

  4. Step 4: The equation of a line with intercepts a and b on the axes is:

    xa+yb=1\frac{x}{a} + \frac{y}{b} = 1

  5. Step 5: Substituting the values of a and b:

    x3h/2+y3k=1\frac{x}{3h/2} + \frac{y}{3k} = 1 2x3h+y3k=1\frac{2x}{3h} + \frac{y}{3k} = 1

  6. Step 6: Multiplying both sides by 3: 2xh+yk=3\frac{2x}{h} + \frac{y}{k} = 3 or 2kx+hy=3hk2kx + hy = 3hk

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