Question

23. Find the ratio in which the join A(2,1,5) and B(3,4,3) is divided by the plane 2x+2y-2z=1
Also find the co-ordinate of the point of division​

Answer 1

Plane P divides the line AB internally in the ratio 5 : 7. The co-ordinates of the point of division are $D = (\frac{29}{12} \ ,\frac{9}{4} \ ,\frac{25}{6} )$

Explanation:

Given:

$A\ (2,1,5)$ ≡ $(x_1,y_1,z_1)$

$B\ (3,4,3)$ ≡ $(x_2,y_2, z_2)$

Plane P = $2x+2y-2z=1$ ......................(1)

Let $D\ (x_3,y_3, z_3)$ be the point at which plane P meets the line segment formed by points A and B.

Let the plane P divide line AB in the ratio $k:1$ ; [m : n = k : 1]

m = k , n = 1

Point D lies on the line AB and also on the plane P. The co-ordinates of D are equal to:

$D = (\frac{mx_2 + nx_1}{m+n}\ ,\frac{my_2 + ny_1}{m+n}\ ,\frac{mz_2 + nz_1}{m+n} )\\\\D = (\frac{3k +2}{k+1}\ ,\frac{4k + 1}{k+1}\ ,\frac{3k + 5}{k+1} )$..........................(2)

Since point D lies on the plane P, it should satisfy equation (1)

Substituting the values of x, y and z co-ordinates of point D in equation of plane P, we get:

$2x+2y-2z=1\\\\2(\frac{3k +2}{k+1})+2(\frac{4k + 1}{k+1})-2(\frac{3k + 5}{k+1})=1\\\\6k+4+8k+2-6k-10=k+1\\8k-4=k+1\\8k-k=4+1\\7k=5\\\\k=\frac{5}{7}$

The required ratio is k : 1 , i.e. $\frac{5}{7} :1$ = 5 : 7

Therefore plane P divides line AB internally in the ratio 5 : 7.

To find the coordinates of point D, substitute the value of k in equation (2)

$D = (\frac{3k +2}{k+1}\ ,\frac{4k + 1}{k+1}\ ,\frac{3k + 5}{k+1} )\\\\D = (\frac{3[\frac{5}{7}]+2}{[\frac{5}{7}]+1}\ ,\frac{4[\frac{5}{7}] + 1}{[\frac{5}{7}]+1}\ ,\frac{3[\frac{5}{7}]+5}{[\frac{5}{7}]+1} )\\\\ D = (\frac{[\frac{15+14}{7}]}{[\frac{5+7}{7}]}\ ,\frac{[\frac{20+7}{7}]}{[\frac{5+7}{7}]}\ ,\frac{[\frac{15+35}{7}]}{[\frac{5+7}{7}]} )\\\\D = (\frac{[\frac{29}{7}]}{[\frac{12}{7}]}\ ,\frac{[\frac{27}{7}]}{[\frac{12}{7}]}\ ,\frac{[\frac{50}{7}]}{[\frac{12}{7}]} )$

$D = (\frac{29}{12} \ ,\frac{27}{12} \ ,\frac{50}{12} )$

$D = (\frac{29}{12} \ ,\frac{9}{4} \ ,\frac{25}{6} )$

Therefore co-ordinates of the point of division D are $D = (\frac{29}{12} \ ,\frac{9}{4} \ ,\frac{25}{6} )$

To learn more about Planes,

https://brainly.in/question/3236112

Find the distance between the planes x+2y-3z=8 and 3x+6y-9z=10

https://brainly.in/question/8252565

Find the ratio in which the plane x-2y+3z=17 divides the line joining points (-2,4,7)&(3,-5,8)