Question

find the value of k....
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Answer 1

Answer :

Diagram :

$(-6,-2) \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: (30,16)$

•-------------•-------------•

$\sf A \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: p \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: B$

Solution :

Here using the section formula, if a point (x,y) divides the line joining the points $(x_1 , y_1) and (x_2, y_2)$ in the ratio m : n = k : 1 = AP : PB

Then,

$\sf (x,y) = \bigg( \frac{mx_2 + nx_1}{m + n} , \frac{my_2 + ny_1}{m + n } \bigg)$

The ratio is k : 1,

And

$x_1 = -6$

$x_2 = 30$

$y_1 = -2$

$y_2 = 16$

Now,

Coordinates Of Point P ( x , y )

$\sf = \bigg( \frac{mx_2+ nx_1}{m +n } , \frac{my_2 + ny_1}{m + n } \bigg)$

m = k and n = 1

$\sf = \bigg( \frac{k(30) + (-6)}{k +1} , \frac{k(16) + 1(-2)}{k + 1} \bigg)$

$\sf = \bigg( \frac{30k - 6 }{k + 1} , \frac{16k - 2 }{k + 1} \bigg)$

Now, as given in question

P lies on the line x + y - 4 = 0

We have , x and y

x = $\sf \bigg( \frac{30k - 6 }{k + 1} \bigg)$

y = $\sf \bigg( \frac{16k - 2 }{k + 1} \bigg)$

Now we have to put this values in this equation : x + y - 4 = 0

x + y - 4 = 0

$\sf \bigg( \frac{30k - 6 }{k + 1} \bigg) + \bigg( \frac{16k - 2 }{k + 1} \bigg) - 4 = 0$

$\sf \bigg( \frac{30k - 6 }{k + 1} \bigg) + \bigg( \frac{16k - 2 }{k + 1} \bigg) = 4$

$\sf \bigg( \frac{30k - 6 + 16k - 2}{k + 1} = 4$

$\sf \bigg( \frac{46k - 8 }{k + 1} = 4$

$\sf 46k - 8 = 4(k +1)$

46k - 8 = 4k + 4

46k - 4k = 8 + 4

42k = 12

k = $\sf \frac{12}{42}$

$\sf \red{\underline{\underline{k = \frac{2}{7}}}}$

I hope it helps you ❤️✔️