Question

find the ratio in which point P(k,7)divides the segment joining A(8,9)and B(1,2).also find k.

Answer 1

Answer:

2:5

k = 6

Step-by-step explanation:

Let the point P(k, 7) divides the segment joining A (8, 9) and B(1,2) in the

ratio λ:1

By using internal sectional formula,

P will be of the form (λ + 8/λ+1, 2λ + 9/λ + 1)

But P is (k, 7) By comparing y-coordinate, we get

2λ + 9/λ + 1 = 7

=> 2λ + 9 = 7λ + 7

=> 5λ = 2

=> λ = 2/5

Hence the ratio in which P divides is 2 :5.

Coordinates of P are (2+5*8/7, 4+5*9/7)

=(6, 7).

Thus, k = 6

Answer 2

Solution:

Let the point P(k,7) divide the segment joining A(8,9)and B(1,2) in m:n,
from section formula we know that coordinate of that point can be calculated as

$x = \frac{mx_{1} + nx_{2}}{m + n} \\ \\ y = \frac{my_{1} + ny_{2}}{m + n} \\ \\ k = \frac{8m + n}{m+ n} ...eq1$
To find the ratio in which the point P divides the line segment joining A and B,if we take that ratio as h:1,than it can easily solved by one coordinate only
So,

$7 = \frac{9h + 2}{h + 1}...eq2 \\ \\ 7h + 7 = 9h + 2 \\ \\ (7 - 9)h= 2 - 7 \\ \\ - 2h = - 5 \\ \\ h = \frac{5}{2}$

so m:n = 5:2

put the value of m and n in eq 1 to get the value of k

$k = \frac{8 \times 5 + 2}{5 + 2} \\ \\ k= \frac{42}{7} \\ \\ k = 6$

value if k = 6

so, p(6,7)

hope it helps you.