Question

If $xy-$plane and $yz-$plane divides the line segment joining $A(2,4,5)$ and $B(3,5,-4)$ in the ratio $a:b$ and $p:q$ respectively then value of $(\frac{a}{b}+\frac{p}{q})$ may be

• A. $\frac{23}{12}$
• B. $\frac{7}{5}$
• C. $\frac{7}{12}$
• D. $\frac{21}{10}$

Answer 1

Answer: (C)

$\frac{7}{12}$

Let the point on $xy$-plane which divide line AB in the ratio a:b be $\left(x_1,y_1,0\right)$ and the point on $yz$-plane which divide line AB in the ratio p:q be $\left(0,y_2,z_2\right)$.

Then $\left(x_1,y_1,0\right)=\left(\frac{3a+2b}{a+b},\frac{5a+4b}{a+b},\frac{-4a+5b}{a+b}\right)$

$\Rightarrow \frac{-4a+5b}{a+b}=0$

$\Rightarrow 4a=5b$

$\Rightarrow \frac{a}{b}=\frac{5}{4}$

and $\left(0,y_2,z_2\right)=\left(\frac{3p+2q}{p+q},\frac{5p+4q}{p+q},\frac{-4p+5q}{p+q}\right)$

$\Rightarrow \frac{3p+2q}{p+q}=0$

$\Rightarrow 3p=-2q$

$\Rightarrow \frac{p}{q}=\frac{-2}{3}$

Now, $\frac{a}{b}+\frac{p}{q}=\frac{5}{4}-\frac{2}{3}=\frac{15-8}{12}=\frac{7}{12}$