Question

Determine the ratio , coordinate geometry question

Answer 1

${\huge {\mathfrak {\orange {Answer :-}}}}$

$<b>$ First, we need to find out the equation for the points A (2, - 2) and B (3,7).

So, we have as follows,

m = (y2 - y1) / (x2 - x1)

= (7 + 2) / (3 - 2)

= 9 / 1 = 9.

Now, we get that,

y = mx + b

y = (9)x + b.

Putting values of x and y from any one of the equations,

y = 9x + b

Or, (7) = 9(3) + b

Or, 7 = 27 + b

Or, b = - 20.

Now, putting the value of b,

y = 9(x) + b

Or, y = 9x - 20. $</b>$

$<i>$ Now, we need to find the intersecting point of the lines.

So, we have the two lines,

2x + y - 4 = 0

Or, 2x + y = 4

Or, y = 4 - 2x

And, we have, y = 9x - 20.

So, putting the values,

4 - 2x = 9x - 20

Or, 11x = 24

Or, x = 24/11

And now, solving for y,

y = 4 - 2x

= 4 - 2(24/11)

= 4 - 48/11

= - 4/11

Thus, the lines intersect each other at the point (24/11, - 4/11).$</i>$

$<b>$ Let the ratio in which the line is divided be m : n.

Thus, we have the coordinates,

= [(mx2 + nx1) / (m + n), (my2 + ny1) / m + n]

= [(3m + 2n) / (m + n) , (7m - 2n) / (m + n) ]

But, we got the coordinates as (24/11, - 4/11)$</b>$

Now, putting the value in any of the coordinates,

(3m + 2n) / (m + n) = 24 / 11

Or, 33m + 22n = 24m + 24n

Or, 9m = 2n

Or, m/n = 2/9

Or, m : n = 2 : 9

➡️ $<u>Hence, the required ratio is m : n, which is 2 : 9 </u>$

$<b>$ Fabulous question..!! Got me working really hard.
$<marquee >$
${\huge {\mathfrak {\orange {That's\: it..}}}}$

${\huge {\mathfrak {\red {-\: Arnab}}}}$