If the mirror image of the point (2,4,7) in the plane 3x-y+4z=2 is (a,b,c), then 2a+b+2c i…

Question

If the mirror image of the point $(2,4,7)$ in the plane $3x-y+4z=2$ is $(a,b,c)$, then $2a+b+2c$ is equal to:

Accepted Answer

We know mirror image of point (x1, y1, z1) in the plane ax + by + cz = d

$${{x - {x_1}} \over a} = {{y - {y_1}} \over b} = {{z - {z_1}} \over c} = {{ - 2(a{x_1} + b{y_1} + c{z_1} - d)} \over {{a^2} + {b^2} + {c^2}}}$$

Here given point (2, 4, 7) and plane $3x - y + 4z = 2$ then mirror image is

$${{x - 2} \over 3} = {{y - 4} \over { - 1}} = {{z - 7} \over 4} = {{ - 2(6 - 4 + 28 - 2)} \over {9 + 1 + 16}}$$

$$ \Rightarrow {{x - 2} \over 3} = {{y - 4} \over { - 1}} = {{z - 7} \over 4} =  - {{28} \over {13}}$$

$$\therefore$$ $$x =  - {{58} \over {13}} = a$$

$$y = {{80} \over {13}} = b$$

$$z =  - {{21} \over {13}} = c$$

$$\therefore$$ $2a + b + 2c$

$$ = 2\left( { - {{58} \over {13}}} \right) + {{80} \over {13}} + 2\left( { - {{21} \over {13}}} \right)$$

$$ = {{ - 116 + 80 - 42} \over {13}} = {{ - 78} \over {13}} =  - 6$$

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