question_answer

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

A)  \[2kx+hy=3hk\]

B)  \[2kx+hy=2hk\]

C)  \[2kx-hy=3hk\]

D)  None of the above

Correct Answer: A

Solution :

Let the equation of line AB is \[\frac{x}{a}+\frac{y}{b}=1\]                       ?.(i) Let a point R (h, k) divide line AB in the ratio 1:2, By using internally ratio, \[R(h,k)=\left( \frac{1{{x}_{2}}+2{{x}_{1}}}{1+2},\frac{1{{y}_{2}}+2{{y}_{1}}}{1+2} \right)\] \[\because \] \[h=\frac{1\times 0+2\times a}{1+2},k=\frac{1\times b+2\times 0}{1+2}\] [\[\because \]P(x, y) divide the line \[A({{x}_{1}},{{y}_{1}})\]and\[B({{x}_{2}},{{y}_{2}})\]in the ratio m:n internally] \[\therefore \]  \[P(x,y)=\left( \frac{n{{x}_{2}}+m{{x}_{2}}}{n+m},\frac{n{{y}_{2}}+m{{y}_{1}}}{n+m} \right)\] \[\Rightarrow \]               \[h=\frac{2a}{3},k=\frac{b}{3}\] \[\Rightarrow \]               \[a=\frac{3h}{2},b=3k\] On putting the values of a and b in Eq. (i), we get                 \[\frac{x}{\left( \frac{3h}{2} \right)}+\frac{y}{3k}=1\Rightarrow \frac{2x}{3h}+\frac{y}{3k}=1\]                 \[\Rightarrow \]               \[\frac{2kx+hy}{3hk}=1\]                                 \[2kx+hy=3hk\]