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Let a, b R. If the mirror image of the point P(a, 6, 9) with respect to the line (x - 3)/7 = (y - 2)/5 = (z - 1)/- 9 (20, b, - a - 9) then |a b| * is | equal to (2021 Main, 24 Feb II) (d) 90 (a) 88 (b) 86 (c) 84 ppo thro the?
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Let a, b R. If the mirror image of the point P(a, 6, 9) with respect t...
Problem Analysis:
- We are given a point P(a, 6, 9) and a line with equation (x - 3)/7 = (y - 2)/5 = (z - 1)/-9.
- We need to find the mirror image of point P with respect to the given line.
- Then, we need to find the value of |a b| *.
Solution:
Finding the Mirror Image:
- The line with equation (x - 3)/7 = (y - 2)/5 = (z - 1)/-9 can be written as:
x = 7t + 3,
y = 5t + 2,
z = -9t + 1,
where t is a parameter.
- Let Q(x, y, z) be the mirror image of point P with respect to the given line.
- To find Q, we can use the formula for reflection of a point about a line.
- The formula states that if the coordinates of a point P are (x₁, y₁, z₁) and the equation of the line is ax + by + cz + d = 0, then the coordinates of the reflection point Q are given by:
Q(x, y, z) = P - 2[(ax₁ + by₁ + cz₁ + d)/(a² + b² + c²)](a, b, c)
- Applying this formula, we have:
Q(x, y, z) = P - 2[(7(a) + 5(b) - 9(c) + (-3)(1))/(7² + 5² + (-9)²)](7, 5, -9)
- Simplifying further, we get:
Q(x, y, z) = P - 2[7a + 5b - 9c - 3]/175(7, 5, -9)
- Substituting the coordinates of P, we get:
Q(x, y, z) = (a, 6, 9) - 2[7a + 5b - 9c - 3]/175(7, 5, -9)
= (a, 6, 9) - 2(7a + 5b - 9c - 3)(7/175, 5/175, -9/175)
= (a, 6, 9) - (2/25)(7a + 5b - 9c - 3)(7, 5, -9)
- Expanding the equation and simplifying, we get:
Q(x, y, z) = (a, 6, 9) - (2/25)(49a + 35b - 63c - 21, 35a + 25b - 45c - 15, -63a - 45b + 81c + 27)
= (a - (98/25)(49a + 35b - 63c - 21), 6 - (70/25)(49a + 35b - 63c - 21), 9 + (126/25)(49a + 35b - 63
- We are given a point P(a, 6, 9) and a line with equation (x - 3)/7 = (y - 2)/5 = (z - 1)/-9.
- We need to find the mirror image of point P with respect to the given line.
- Then, we need to find the value of |a b| *.
Solution:
Finding the Mirror Image:
- The line with equation (x - 3)/7 = (y - 2)/5 = (z - 1)/-9 can be written as:
x = 7t + 3,
y = 5t + 2,
z = -9t + 1,
where t is a parameter.
- Let Q(x, y, z) be the mirror image of point P with respect to the given line.
- To find Q, we can use the formula for reflection of a point about a line.
- The formula states that if the coordinates of a point P are (x₁, y₁, z₁) and the equation of the line is ax + by + cz + d = 0, then the coordinates of the reflection point Q are given by:
Q(x, y, z) = P - 2[(ax₁ + by₁ + cz₁ + d)/(a² + b² + c²)](a, b, c)
- Applying this formula, we have:
Q(x, y, z) = P - 2[(7(a) + 5(b) - 9(c) + (-3)(1))/(7² + 5² + (-9)²)](7, 5, -9)
- Simplifying further, we get:
Q(x, y, z) = P - 2[7a + 5b - 9c - 3]/175(7, 5, -9)
- Substituting the coordinates of P, we get:
Q(x, y, z) = (a, 6, 9) - 2[7a + 5b - 9c - 3]/175(7, 5, -9)
= (a, 6, 9) - 2(7a + 5b - 9c - 3)(7/175, 5/175, -9/175)
= (a, 6, 9) - (2/25)(7a + 5b - 9c - 3)(7, 5, -9)
- Expanding the equation and simplifying, we get:
Q(x, y, z) = (a, 6, 9) - (2/25)(49a + 35b - 63c - 21, 35a + 25b - 45c - 15, -63a - 45b + 81c + 27)
= (a - (98/25)(49a + 35b - 63c - 21), 6 - (70/25)(49a + 35b - 63c - 21), 9 + (126/25)(49a + 35b - 63
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Let a, b R. If the mirror image of the point P(a, 6, 9) with respect to the line (x - 3)/7 = (y - 2)/5 = (z - 1)/- 9 (20, b, - a - 9) then |a b| * is | equal to (2021 Main, 24 Feb II) (d) 90 (a) 88 (b) 86 (c) 84 ppo thro the?
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Let a, b R. If the mirror image of the point P(a, 6, 9) with respect to the line (x - 3)/7 = (y - 2)/5 = (z - 1)/- 9 (20, b, - a - 9) then |a b| * is | equal to (2021 Main, 24 Feb II) (d) 90 (a) 88 (b) 86 (c) 84 ppo thro the? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let a, b R. If the mirror image of the point P(a, 6, 9) with respect to the line (x - 3)/7 = (y - 2)/5 = (z - 1)/- 9 (20, b, - a - 9) then |a b| * is | equal to (2021 Main, 24 Feb II) (d) 90 (a) 88 (b) 86 (c) 84 ppo thro the? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let a, b R. If the mirror image of the point P(a, 6, 9) with respect to the line (x - 3)/7 = (y - 2)/5 = (z - 1)/- 9 (20, b, - a - 9) then |a b| * is | equal to (2021 Main, 24 Feb II) (d) 90 (a) 88 (b) 86 (c) 84 ppo thro the?.
Let a, b R. If the mirror image of the point P(a, 6, 9) with respect to the line (x - 3)/7 = (y - 2)/5 = (z - 1)/- 9 (20, b, - a - 9) then |a b| * is | equal to (2021 Main, 24 Feb II) (d) 90 (a) 88 (b) 86 (c) 84 ppo thro the? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let a, b R. If the mirror image of the point P(a, 6, 9) with respect to the line (x - 3)/7 = (y - 2)/5 = (z - 1)/- 9 (20, b, - a - 9) then |a b| * is | equal to (2021 Main, 24 Feb II) (d) 90 (a) 88 (b) 86 (c) 84 ppo thro the? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let a, b R. If the mirror image of the point P(a, 6, 9) with respect to the line (x - 3)/7 = (y - 2)/5 = (z - 1)/- 9 (20, b, - a - 9) then |a b| * is | equal to (2021 Main, 24 Feb II) (d) 90 (a) 88 (b) 86 (c) 84 ppo thro the?.
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Let a, b R. If the mirror image of the point P(a, 6, 9) with respect to the line (x - 3)/7 = (y - 2)/5 = (z - 1)/- 9 (20, b, - a - 9) then |a b| * is | equal to (2021 Main, 24 Feb II) (d) 90 (a) 88 (b) 86 (c) 84 ppo thro the?, a detailed solution for Let a, b R. If the mirror image of the point P(a, 6, 9) with respect to the line (x - 3)/7 = (y - 2)/5 = (z - 1)/- 9 (20, b, - a - 9) then |a b| * is | equal to (2021 Main, 24 Feb II) (d) 90 (a) 88 (b) 86 (c) 84 ppo thro the? has been provided alongside types of Let a, b R. If the mirror image of the point P(a, 6, 9) with respect to the line (x - 3)/7 = (y - 2)/5 = (z - 1)/- 9 (20, b, - a - 9) then |a b| * is | equal to (2021 Main, 24 Feb II) (d) 90 (a) 88 (b) 86 (c) 84 ppo thro the? theory, EduRev gives you an
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