Answer to Question #139820 in Analytic Geometry for Dhruv Rawat

Question

The line x+1=1-y=-5z lies in the plane 2x +3y -5z =1.
True or false with full explanation

Answer 1

Given ,

Equation of line

$x+1=1-y=-5z$

or $\dfrac{x+1}{1}=\dfrac{y-1}{-1}=\dfrac{z}{-\frac{1}{5}}$

$\therefore$ Direction ratios of line are (1,-1,$\dfrac{-1}{5})$

Equation of plane

2x+3y-5z=1

$\therefore$ Direction ratios of plane are (2,3,-5)

Multiplying the Direction ratio of plane and line adding them

$\to1\times2-1\times3+\dfrac{-1}{5}\times(-5)$

=$2-3+1=0$

hence the Product and sum of direction ratio of line and plane is 0.

So Line is parallel to the plane

Also, The distance of the plane from the line is given by the formula,

$d=\dfrac{|2x_1+3y_1-5z_1-1|}{\sqrt{2^2+3^2+(-5)^2}}$

Since the given line passes through (-1,1,0)

$\to d=\dfrac{|-2+3-0-1|}{\sqrt{4+9+25}}$

$\to d=0$

Since The distance between line and plane is 0 and line is parallel to the plane.

So the line must lie in a plane.

Hence It is true Given Line lies in a plane.