Perpendicular Lines MCQ Quiz in मल्याळम - Objective Question with Answer for Perpendicular Lines - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Calculation

Direction ratios of line RQ:

(2-0, -3-(-1), -1-3) = (2, -2, -4) = (1, -1, -2)

Equation of line RQ in parametric form:

Let the foot of the perpendicular from P on line RQ be F(, -1-, 3-2).

Direction ratios of PF:

(-1, -1--8, 3-2-4) = (-1, -9-, -1-2)

Since PF is perpendicular to RQ, the dot product of their direction ratios is zero:

⇒ 1(-1) - 1(-9-) - 2(-1-2) = 0

⇒  - 1 + 9 + + 2 + 4 = 0

⇒ 6 + 10 = 0

⇒ 6 = -10

⇒  = -10/6 = -5/3

Coordinates of F:

∴ The coordinates of the foot of the perpendicular are (-5/3, 2/3, 19/3).

Hence option 3 is correct

Concept:

Then invariant point of a transformation w = T(z) is given by z = T(z).

Explanation:

The invariant points of  is given by

⇒ Z² + 2Z = 2Z - 4

⇒ Z² = -4

⇒ Z = ± 2i

(3) is true.

Let (x,y) be foot of perpendicular drawn to the point on the line

Relation :

Here =(0,0)

given line is:

and

Hence foot of perpendicular

Line meets X-axis at

and meets Y-axis at

Hence, correct option is 'A'.

Equation of is
....(1)
Equation of is
...(2)
By (1) and (2)

Now,

Length of perpendicular

, which is greater than but

Slope of given line is

Lines are perpendicular so

Calculation

Given 

x = y, z = 1 ⇒ L₁ : 

⇒ Q = (λ, λ, 1)

⇒ 

 ⇒   = 0

x = –y, z = –1 ⇒ L₂ : 

⇒ R = (k, -k, -1)

⇒ 

 ⇒  = 0

⇒ k - a + k + a  = 0 ⇒ k = 0

∠QPR is a right angle ⇒ 

⇒

⇒ 

⇒ (1-a)(1+a) = 0 ⇒ a² = 1

⇒12a2 = 12

Concept:

Let two lines having direction ratio’s a_1, b₁, c₁ and a_2, b₂, c₂ respectively.

Condition for perpendicular lines: a₁a₂ + b₁b₂ + c₁c₂ = 0

Calculation:

Direction ratio’s of two lines are given as 1, 2, p − 1 and -3, 1, 2

Lines are perpendicular,

∴ 1 × -3 + 2 × 1 + (p – 1) × 2 = 0

⇒ -3 + 2 + 2p – 2 = 0

⇒ 2p = 3

∴ p = 3/2

Concept:

Let the two lines have direction ratio’s a1, b1, c1 and a2, b2, c2 respectively.

Condition for perpendicular lines: a1a2 + b1b2 + c1c2 = 0

Calculation:

Given lines are   and  

Write the above equation of a line in the standard form of lines

So, the direction ratio of the first line is (2, 2, k)

So, direction ratio of second line is (-k, 2, 5)

Lines are perpendicular,

∴ (2 × -k) + (2 × 2) + (k × 5) = 0

⇒ -2k + 4 + 5k = 0

⇒ 3k + 4 = 0

∴ k = -4/3

Concept:

1. Equation of a line: The equation of a line with direction ratio (a, b, c) that passes through the point (x₁, y₁, z₁) is given by the formula: 

2. Let two lines having direction ratio’s a_1, b₁, c₁ and a_2, b₂, c₂ respectively.

Condition for perpendicular lines: a₁a₂ + b₁b₂ + c₁c₂ = 0

Note: Direction ratio’s of x-axis, y-axis and z-axis are (1, 0, 0), (0, 1, 0) and (0, 0, 1) respectively.

Calculation:

Given line is  

So, Direction ratio’s of the line is (1, 0, 5)

As we know that direction ratio’s of the y-axis is (0, 1, 0)

Now, apply the condition of perpendicular lines,

⇒ 1 × 0 + 0 × 1 + 5 × 0 = 0

Hence, y−axis and given line are perpendicular to each other.