Question

The area of the figure formed by the points(−1,−1,1);(1,1,1)and their mirror images on the plane3x+2y+4z+1=03x+2y+4z+1=0 is
(A) 533√2953329

Answer 1

Answer:

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Step-by-step explanation:

Points of quadrilateral = (-1,-1,1) and (1,1,1)

Equation of plane = 3x + 2y + 4z +1 = 0

To find :

Area of figure formed by points and their image with respect to given plane.

Solution :

To get the area of this figure we firstly have to find the other two points which are the mirror images of given two points with respect to given plane . after finding these two points we can find the area of the figure.

Formula to find the mirror points with respect to given plane ( ax + by + cz + d = 0) :

(equation 1)

For points A (-1,-1,1) and plane 3x + 2y + 4z +1 = 0

here (a,b,c,d) = (3,2,4,1)

The mirror points will be :

on solving this, we get point C :

Similarly for points B (1,1,1) on plane 3x + 2y + 4z +1 = 0

Putting in equation 1, we get

on solving this, we get point D :

To find the area of quadrilateral,

we will choose two sides adjacent to each other,

so we will find AB as

and AC as,

so,

Area of quadrilateral :

(equation 2)

So, Area of quadrilateral is the magnitude of cross product of two adjacent sides of quadrilateral

By putting the value of AB and AC in equation 2,

So,

Cross product of AB and AC is :

So,

Area of quadrilateral is the magnitude of this cross product :