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 :

<p>We know mirror image of point (x₁, y₁, z₁) in the plane ax + by + cz = d</p> <p>$${{x - {x_1}} \over a} = {{y - {y_1}} \over b} = {{z - {z_1}} \over c} = {{ - 2(a{x_1} + b{y_1} + c{z_1} - d)} \over {{a^2} + {b^2} + {c^2}}}$$</p> <p>Here given point (2, 4, 7) and plane $3x - y + 4z = 2$ then mirror image is</p> <p>$${{x - 2} \over 3} = {{y - 4} \over { - 1}} = {{z - 7} \over 4} = {{ - 2(6 - 4 + 28 - 2)} \over {9 + 1 + 16}}$$</p> <p>$$ \Rightarrow {{x - 2} \over 3} = {{y - 4} \over { - 1}} = {{z - 7} \over 4} = - {{28} \over {13}}$$</p> <p>$\therefore$ $x = - {{58} \over {13}} = a$</p> <p>$y = {{80} \over {13}} = b$</p> <p>$z = - {{21} \over {13}} = c$</p> <p>$\therefore$ $2a + b + 2c$</p> <p>$$ = 2\left( { - {{58} \over {13}}} \right) + {{80} \over {13}} + 2\left( { - {{21} \over {13}}} \right)$$</p> <p>$= {{ - 116 + 80 - 42} \over {13}} = {{ - 78} \over {13}} = - 6$</p>