SATHEE: 3D Geometry 3 Question 21

3D Geometry 3 Question 21

21. The direction ratios of normal to the plane through the points $(0, - 1, 0)$ and $(0, 0, 1)$ and making an angle $π / 4$ with the plane $y - z + 5 = 0$ are

(2019 Main, 11 Jan I)

(a) $2, - 1, 1$

(b) $\sqrt{2}, 1, - 1$

(d) $2 \sqrt{3}, 1, - 1$

Show Answer

Answer:

(b,c)

Solution:

Let the equation of plane be

$a (x - 0) + b (y + 1) + c (z - 0) = 0$

$[∵$ Equation of plane passing through a point $(x_{1}, y_{1}, z_{1})$ is given by $a (x - x_{1}) + b (y - y_{1}) + c (z - z_{1}) = 0]$ $\Rightarrow a x + b y + c z + b = 0$

Since, it also passes through $(0, 0, 1)$ , therefore, we get

$c + b = 0$

Now, as angle between the planes

$a x + b y + c z + b = 0$

and

$y - z + 5 = 0 is \frac{π}{4}$

$∴ \cos \frac{π}{4} = \frac{| n_{1} \cdot n_{2} |}{| n_{1} | | n_{2} |};$ where $n_{1} = a \hat{i} + b \hat{j} + c \hat{k}$

and $n_{2} = 0 \hat{i} + \hat{j} - \hat{k}$

$\begin{aligned} \Rightarrow \frac{1}{\sqrt{2}} & = \frac{| (a \hat{i} + b \hat{j} + c \hat{k}) \cdot (0 \hat{i} + \hat{j} - \hat{k}) |}{\sqrt{a^{2} + b^{2} + c^{2}} \sqrt{0 + 1 + 1}} \\ = \frac{| b - c |}{\sqrt{a^{2} + b^{2} + c^{2}} \sqrt{2}} \end{aligned}$

$\Rightarrow a^{2} + b^{2} + c^{2} = | b - c |^{2} = (b - c)^{2} = b^{2} + c^{2} - 2 b c$

$\Rightarrow a^{2} = - 2 b c$

$\Rightarrow a^{2} = 2 b^{2}$

$\Rightarrow a = \pm \sqrt{2} b$

[Using Eq. (ii)]

$\Rightarrow$ Direction ratios $(a, b, c) = (\pm \sqrt{2}, 1, - 1)$

So, options (b) and (c) are correct because

$2, \sqrt{2}, - \sqrt{2}$ and $\sqrt{2}, 1, - 1$ . are multiple of each other.

पिछला

अगला

3D Geometry 3 Question 21

21. The direction ratios of normal to the plane through the points $(0, - 1, 0)$ and $(0, 0, 1)$ and making an angle $π / 4$ with the plane $y - z + 5 = 0$ are

Answer:

Solution:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

3D Geometry 3 Question 21

21. The direction ratios of normal to the plane through the points (0,−1,0) and (0,0,1) and making an angle π/4 with the plane y−z+5=0 are

Answer:

Solution:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Menu

21. The direction ratios of normal to the plane through the points $(0, - 1, 0)$ and $(0, 0, 1)$ and making an angle $π / 4$ with the plane $y - z + 5 = 0$ are