The acute angle between the planes P1 and P2, when P1 and P2 are the planes passing through the intersection of the planes 5x + 8y + 13z – 29 = 0 and 8x – 7y + z – 20 = 0 and the points (2, 1, 3) and (0, 1, 2), respectively, is

( 5 x + 8 y + 1 3 z − 2 9 ) + λ ( 8 x − 7 y + z − 2 0 ) = 0 P1 passing (2, 1, 3)(10 + 8 + 39 – 29) + λ ( 1 6 − 7 + 3 − 2 0 ) = 0  2 8 − 8 λ = 0 λ = 7 / 2  ⇒ 2X – Y + Z – 6 = 0 ….(i)For P2 passes (0, 1, 2) ( 8 + 2 6 − 2 9 ) + λ ( − 7 + 2 − 2 0 ) = 0  5 − 2 5 λ = 0 ⇒ λ = 1 5  P 2 : x + y + 2 z − 5 = 0 . . . . . . . . ( i i )  Acute angle between the planes c o s θ = 2 − 1 + 2 4 + 1 + 1 1 + 1 + 4 = 1 2  ⇒ θ = π 3

The acute angle between the planes P1 and P2, when P1 and P2 are the planes passing through the intersection of the planes 5x + 8y + 13z – 29 = 0 and 8x – 7y + z – 20 = 0 and the points (2, 1, 3) and (0, 1, 2), respectively, is

Ask & Answer Question

Maths Class 12th Three Dimensional Geometry Maths NCERT Exemplar Solutions Class 12th Chapter Ten

The acute angle between the planes P₁ and P₂, when P₁ and P₂ are the planes passing through the intersection of the planes 5x + 8y + 13z – 29 = 0 and 8x – 7y + z – 20 = 0 and the points (2, 1, 3) and (0, 1, 2), respectively, is

Option 1 -

$\frac{π}{3}$

Option 2 -

$\frac{π}{4}$

Option 3 -

$\frac{π}{6}$

Option 4 -

$\frac{π}{12}$

2 Views | Posted 2 months ago

Asked by Shiksha User

1 Answer

Answered by

Payal Gupta | Contributor-Level 10

2 months ago

Correct Option - 1

Detailed Solution:

$(5 x + 8 y + 13 z - 29) + λ (8 x - 7 y + z - 20) = 0$

P₁ passing (2, 1, 3)

(10 + 8 + 39 – 29) + $λ (16 - 7 + 3 - 20) = 0$

$28 - 8 λ = 0 λ = 7 / 2$

$\Rightarrow$ 2X – Y + Z – 6 = 0 ….(i)

For P₂ passes (0, 1, 2)

$(8 + 26 - 29) + λ (- 7 + 2 - 20) = 0$

$5 - 25 λ = 0 \Rightarrow λ = \frac{1}{5}$

$P_{2} : x + y + 2 z - 5 = 0 . . . . . . . . (i i)$

Acute angle between the planes

$c o s θ = \frac{2 - 1 + 2}{\sqrt[]{4 + 1 + 1} \sqrt[]{1 + 1 + 4}} = \frac{1}{2}$

$\Rightarrow θ = \frac{π}{3}$

<math> <mrow> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mi>x</mi> <mo>+</mo> <mn>8</mn> <mi>y</mi> <mo>+</mo> <mn>1</mn> <mn>3</mn> <mi>z</mi> <mo>−</mo> <mn>2</mn> <mn>9</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>λ</mi> <mrow> <mo>(</mo> <mrow> <mn>8</mn> <mi>x</mi> <mo>−</mo> <mn>7</mn> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>−</mo> <mn>2</mn> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math>  P1 passing (2, 1, 3)(10 + 8 + 39 – 29) + <math> <mrow> <mi>λ</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mn>6</mn> <mo>−</mo> <mn>7</mn> <mo>+</mo> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math>   <math> <mrow> <mn>2</mn> <mn>8</mn> <mo>−</mo> <mn>8</mn> <mi>λ</mi> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mi>λ</mi> <mo>=</mo> <mn>7</mn> <mo>/</mo> <mn>2</mn> </mrow> </math>                <math> <mrow> <mo>⇒</mo> </mrow> </math> 2X – Y + Z – 6 = 0      ….(i)For P2 passes (0, 1, 2) <math> <mrow> <mrow> <mo>(</mo> <mrow> <mn>8</mn> <mo>+</mo> <mn>2</mn> <mn>6</mn> <mo>−</mo> <mn>2</mn> <mn>9</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>λ</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>7</mn> <mo>+</mo> <mn>2</mn> <mo>−</mo> <mn>2</mn> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math>               <math> <mrow> <mn>5</mn> <mo>−</mo> <mn>2</mn> <mn>5</mn> <mi>λ</mi> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>⇒</mo> <mi>λ</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> </mrow> </math>                <math> <mrow> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mn>2</mn> <mi>z</mi> <mo>−</mo> <mn>5</mn> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </math>               Acute angle between the planes <math> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> </mrow> <mrow> <mroot> <mrow> <mn>4</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> </mrow> <mrow></mrow> </mroot> <mroot> <mrow> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>4</mn> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>                <math> <mrow> <mo>⇒</mo> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mi>π</mi> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> </mrow> </math>

0 Upvotes 0 Downvotes

Similar Questions for you

If $\int_{}^{} (\frac{x^{2} c o s x}{1 + π^{x}} + \frac{1 + s i n^{2} x}{1 + e^{{(s i n x)}^{2023}}}) d x = \frac{π}{4} (π + α) - 2$

Then the value of 'α' is equal to

alok kumar singh

$\int_{- \frac{π}{2}}^{\frac{π}{2}} (\frac{x^{2} c o s x}{1 + π^{2}} + \frac{1 + s i n^{2} x}{1 + e^{{(s i n x)}^{2023}}}) d x = \frac{π}{4} (π + α) - 2$

$\int_{0}^{\frac{π}{2}} {(\frac{x^{2} c o s x}{1 + π^{x}} + \frac{1 + s i n^{2} x}{1 + e^{{(s i n x)}^{2023}}}) + (\frac{x^{2} c o s x}{1 + π^{- x}} + \frac{1 + s i n^{2} x}{1 + e^{- {(s i n x)}^{2023}}})} d x$

$= \frac{π}{4} (π + α) - 2$

$\int_{0}^{\frac{π}{2}} (x^{2} c o s x + 1 + s i n^{2} x) d x = \frac{π}{4} (π + α) - 2$

$\int_{0}^{\frac{π}{2}} x^{2} c o s x d x + \int_{0}^{\frac{π}{2}} (1 + s i n^{2} x) d x = \frac{π}{4} (π + α) - 2$ ....(1)

Let $I_{1} = \int_{0}^{\frac{π}{2}} (1 + s i n^{2} x) d x$

$I_{1} = \int_{0}^{\frac{π}{2}} 1 \cdot d x + \int_{0}^{\frac{π}{2}} (\frac{1 - c o s 2 x}{2}) d x$

$I_{1} = \frac{π}{2} + \frac{1}{2} [\frac{π}{2} + 0]$

$I_{1} = \frac{3 π}{4}$

Let $I_{2} = \int_{0}^{\frac{π}{2}} x^{2} c o s x d x$

$I_{2} = {[x^{2} (s i n x) - \int 2 x \int c o s x d x]}_{0}^{\frac{π}{2}}$

$I_{2} = {[x^{2} (s i n x) - 2 \int x s i n x]}_{0}^{\frac{π}{2}}$

$I_{2} = {[x^{2} s i n x - 2 (x (- c o s x) + \int c o s x)]}_{0}^{\frac{π}{2}}$

$I_{2} = {[x^{2} s i n x - 2 (- x c o s x + s i n x)]}_{0}^{\frac{π}{2}}$

$I_{2} = (\frac{π^{2}}{4} - 2)$

Put l₁ and l₂ in (1)

$\frac{π^{2}}{4} - 2 + \frac{3 π}{4}$

$\frac{π^{2}}{4} + \frac{3 π}{4} - 2$

$\frac{π}{4} (π + 3) - 2$

α = 3

If $| \vec{a} | = 1$ , $| \vec{b} | = 4$ $\vec{a} \cdot \vec{b} = 2$ and $\vec{c} = 2 (\vec{a} \times \vec{b}) - 3 \vec{b}$ Then the angle between $\vec{b}$ and $\vec{c}$ is

alok kumar singh

Given $| \vec{a} | = 1$ , $| \vec{b} | = 4$ , $\vec{a} \cdot \vec{b} = 2$

$\vec{c} = 2 (\vec{a} \times \vec{b}) - 3 \vec{b}$

Dot product with $\vec{a}$ on both sides

$\vec{c} \cdot \vec{a} = - 6$ ... (1)

Dot product with $\vec{b}$ on both sides

$\vec{b} \cdot \vec{c} = - 48$ ... (2)

$\vec{c} \cdot \vec{c} = 4 {| \vec{a} \times \vec{b} |}^{2} + 9 {| \vec{b} |}^{2}$

${| \vec{c} |}^{2} = 4 [{| \vec{a} |}^{2} \cdot {| \vec{b} |}^{2} - {(\vec{a} \cdot \vec{b})}^{2}] + 9 {| \vec{b} |}^{2}$

${| \vec{c} |}^{2} = 4 [(1) {(4)}^{2} - (4)] + 9 (16)$

${| \vec{c} |}^{2} = 4 [12] + 144$

${| \vec{c} |}^{2} = 48 + 144$

${| \vec{c} |}^{2} = 192$

$c o s θ = \frac{\vec{b} \cdot \vec{c}}{| \vec{b} | | \vec{c} |}$

$c o s θ = \frac{- 48}{\sqrt[]{192} \cdot 4}$

$c o s θ = \frac{- 48}{8 \sqrt[]{3} \cdot 4}$

$c o s θ = \frac{- 3}{2 \sqrt[]{3}}$

$c o s θ = \frac{- \sqrt[]{3}}{2}$ $θ = c o s^{- 1} (\frac{- \sqrt[]{3}}{2})$

If the foot of perpendicular from (1, 2, 3) to the line $\frac{x + 1}{2} = \frac{y - 2}{5} = \frac{z - 4}{1}$ is (a, b, g) then find a + b + g

alok kumar singh

(a – 1) × 2 + (b – 2) × 5 + (g – 3) × 1 = 0

2a + 5b + g – 15 = 0

Also, P lie on line

a + 1 = 2λ

b – 2 = 5λ

g – 4 = λ

2 (2λ – 1) + 5 (5λ + 2) + λ + 4 – 15 = 0

4λ + 25λ + λ – 2 + 10 + 4 – 15 = 0

30λ – 3 = 0

$λ = \frac{1}{10}$

a + b + g = (2λ – 1) + (5λ + 2) + (λ + 4)

$= 8 λ + 5 = \frac{8}{10} + 5 = 5.8$

If (α, β, γ) is mirror image of the point (2, 3, 4) with respect to the line $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ Then 2α + 3β + 4γ is

alok kumar singh

Take $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} = λ$

x = 2λ + 1, y = 3λ + 2, z = 4λ + 3

$\vec{A B}$ = (α − 2) $\hat{i}$ + (β − 3) $\hat{j}$ + (γ − 4) $\hat{k}$

Now,

(α − 2) ⋅ 2 + (β − 3) ⋅3 + (γ − 4) ⋅ 4 = 0

2α − 4 + 3β − 9 + 4γ −16 = 0

⇒ 2α + 3β + 4γ = 29

Let $λ$ be an integer. IF the shortest distance between the lines $x - λ = 2 y - 1 = - 2 z$ and $x = y + 2 λ = z - λ i s \frac{\sqrt[]{7}}{2 \sqrt[]{2}}$ , then the value of $| λ | i s . . . . . . . . . . . . . . . . . . .$

Vishal Baghel

$L_{1} = \frac{x - λ}{1} = \frac{y - \frac{1}{2}}{\frac{1}{2}} = \frac{z}{- \frac{1}{2}}$

$∴ S D = \frac{| - 2 λ + 3 (2 λ + \frac{1}{2}) + λ |}{\sqrt[]{14}} = \frac{| 5 λ + \frac{3}{2} |}{\sqrt[]{14}}$

$5 λ + \frac{3}{2} = - \frac{7}{2} \Rightarrow 5 λ = - 5 \Rightarrow λ = - 1$

$∴ | λ | = 1$

Get authentic answers from experts, students and alumni that you won't find anywhere else

On Shiksha, get access to

65k Colleges
1.2k Exams
688k Reviews
1800k Answers

Community GuidelinesUser Points System

QuestionsDiscussionsTags

The acute angle between the planes P₁ and P₂, when P₁ and P₂ are the planes passing through the intersection of the planes 5x + 8y + 13z – 29 = 0 and 8x – 7y + z – 20 = 0 and the points (2, 1, 3) and (0, 1, 2), respectively, is

1 Answer

Similar Questions for you

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question