Question

If a(2,-1) and b(6,5) are two points the ratio in which the foot of perpendicular from (4,1) to ab divides it is

Answer 1

Answer:

please see! the attachment

Answer 2

Answer:

5:8

Step-by-step explanation:

Let us assume that the point d(h,k) divides the line joining a(2,-1) and b(6,5) in the ratio m:n internally and c(4,1) is another point such that cd⊥ab.

Now, from the above condition, d(h,k)≡($\frac{6m+2n}{m+n},\frac{5m-n}{m+n}$)......... (1)

[Here, we applied the formula that the coordinates of R will be

$[\frac{mx_{2}+nx_{1}  }{m+n},\frac{my_{2}+ny_{1}  }{m+n}]$

where point R divides the line joining P($x_{1},y_{1}$) and Q($y_{1},y_{2}$) in the ratio m:n internally]

Now, the slope of ab, $M_{1}$ =$\frac{5-(-1)}{6-2}$=$\frac{3}{2}$......... (2)

Again, the slope of cd,$M_{2}$= $\frac{\frac{5m-n}{m+n}-1}{\frac{6m+2n}{m+n}-4 }$

⇒$M_{2}$=$\frac{4m-2n}{2m-2n}$

⇒$M_{2}$= $\frac{2m-n}{m-n}$....... (3)

From equation (2) and (3), $M_{1} M_{2}=-1$ {Since product of two straight lines perpendicular to each other is -1}

⇒$\frac{3}{2}.\frac{2m-n}{m-n}=-1$

⇒6m-3n=-2m+2n

⇒8m=5n

⇒$\frac{m}{n}=\frac{5}{8}$

Therefore, the ratio m:n = 5:8 (Answer)