Question

the line joining the points 2, -3,1 and 1,2,-4 cuts the plane 2x+3y-5z+3 = 0 at (a,b,c) then the value of a is​

Answer 1

Step-by-step explanation:

Given:

The line joining the points 2,-3,1 and (1,2,-4) cuts the plane 2X+3y-5z+3=0 at (a,b,c).

To find: Value of a

Solution:

Formula to find equation of line passing through two given points A(x1,y1,z1) and B(x2,y2,z2)

$\bold{\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}}$

Here points are

A(2,-3,1), B(1,2,-4)

$\frac{x - 2}{1 - 2} = \frac{y + 3}{2 + 3} = \frac{z - 1}{ - 4 - 1} \\ \\ \frac{x - 2}{ - 1} = \frac{y + 3}{5} = \frac{z - 1}{ - 5}$

Let (x,y,z) are the points where line cuts the plane

$\frac{x - 2}{ - 1} = \frac{y + 3}{5} = \frac{z - 1}{ - 5} = k \\ \\ x = - k + 2 \\ \\ y = 5k - 3 \\ \\ z = - 5k + 1$

these must satisfy equation of plane

put these into plane

$2x + 3y - 5z + 3 = 0 \\ \\ 2( - k + 2) + 3(5k - 3) - 5( - 5k + 1) + 3 = 0 \\ \\ - 2k + 4 + 15k - 9 + 25k - 5 + 3 = 0 \\ \\ 38k - 7 = 0 \\ \\ k = \frac{7}{38}$

Put value of k in coordinates of point of intersection.

$x = - k + 2 \\ \\ x = \frac{ - 7}{38} + 2 = \frac{69}{8} \\ \\ y = 5k - 3 \\ \\ y = 5 \times \frac{7}{38} - 3 \\ \\ y = \frac{35 - 114}{8} = \frac{79}{38} \\ \\ z = - 5k + 1 \\ \\ z = - 5 \times \frac{7}{38} + 1 \\ \\ = \frac{ - 35 + 38}{38} \\ \\ z = \frac{ 3}{38}$

Coordinates of intersection of plane are

69/38,79/38,3/38

ATQ,these are (a,b,c)

Value of a is 69/38

Final answer:

Value of a is 69/38

Hope it helps you.

To learn more on brainly:

1)Write the coordinates of the point where the graph of the equation x +2y = 4 intersect the x-axis

https://brainly.in/question/12816889

2)Find the intercepts on the coordinate axis by the plane 2x-3y+5z=4

https://brainly.in/question/7554997