JEE MAIN - Mathematics (2021 - 24th February Evening Shift - No. 3)
Let a, b$$ \in $$R. If the mirror image of the point P(a, 6, 9) with respect to the line
$${{x - 3} \over 7} = {{y - 2} \over 5} = {{z - 1} \over { - 9}}$$ is (20, b, $$-$$a$$-$$9), then | a + b |, is equal to :
$${{x - 3} \over 7} = {{y - 2} \over 5} = {{z - 1} \over { - 9}}$$ is (20, b, $$-$$a$$-$$9), then | a + b |, is equal to :
88
90
86
84
Explanation
Given, P(a, 6, 9)
Equation of line $${{x - 3} \over 7} = {{y - 2} \over 5} = {{z - 1} \over { - 9}}$$
Image of point P with respect to line is point Q(20, b, $$-$$a $$-$$9)
Mid-point of P and Q = $$\left( {{{a + 20} \over 2},{{6 + b} \over 2},{{ - a} \over 2}} \right)$$
This point lies on line
$$\therefore$$ $${{{{a + 20} \over 2} - 3} \over 7} = {{{{6 + b} \over 2} - 2} \over 5} = {{{{ - a} \over 2} - 1} \over { - 9}}$$
$$ \Rightarrow {{a + 14} \over {14}} = {{b + 2} \over {10}} = {{a + 2} \over {18}}$$
$$ \Rightarrow {{a + 14} \over {14}} = {{a + 2} \over {18}}$$ and $${{b + 2} \over {10}} = {{a + 2} \over {18}}$$
Solving, we get a = $$-$$ 56, b = $$-$$ 32
$$\therefore$$ $$\left| {a + b} \right| = \left| { - 56 - 32} \right| = 88$$
Equation of line $${{x - 3} \over 7} = {{y - 2} \over 5} = {{z - 1} \over { - 9}}$$
Image of point P with respect to line is point Q(20, b, $$-$$a $$-$$9)
Mid-point of P and Q = $$\left( {{{a + 20} \over 2},{{6 + b} \over 2},{{ - a} \over 2}} \right)$$
This point lies on line
$$\therefore$$ $${{{{a + 20} \over 2} - 3} \over 7} = {{{{6 + b} \over 2} - 2} \over 5} = {{{{ - a} \over 2} - 1} \over { - 9}}$$
$$ \Rightarrow {{a + 14} \over {14}} = {{b + 2} \over {10}} = {{a + 2} \over {18}}$$
$$ \Rightarrow {{a + 14} \over {14}} = {{a + 2} \over {18}}$$ and $${{b + 2} \over {10}} = {{a + 2} \over {18}}$$
Solving, we get a = $$-$$ 56, b = $$-$$ 32
$$\therefore$$ $$\left| {a + b} \right| = \left| { - 56 - 32} \right| = 88$$
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