JEE Mains · Maths · STD 12 - 11. three dimension geometry
A line with direction ratios \(1, -1, 2\) intersects the lines \(\dfrac{x}{2} = \dfrac{y}{3} = \dfrac{z+1}{3}\) and \(\dfrac{x+1}{-1} = \dfrac{y-2}{1} = \dfrac{z}{4}\) at the points \(P\) and \(Q\), respectively. If the length of the line segment \(PQ\) is \(\alpha\), then \(225\alpha^2\) is equal to:
- A \(1024\)
- B \(1014\)
- C \(1104\)
- D \(1204\)
Answer & Solution
Correct Answer
(B) \(1014\)
Step-by-step Solution
Detailed explanation
Let the coordinates of \(P\) on the first line be \((2\lambda, 3\lambda, 3\lambda - 1)\). Let the coordinates of \(Q\) on the second line be \((-\mu - 1, \mu + 2, 4\mu)\). The direction ratios of the line segment \(PQ\) are proportional to \(1, -1, 2\). Thus, we can write:…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- \(\mathop {\lim }\limits_{x \to \pi /4} \frac{{{{\cot }^3}\,x - \tan \,x}}{{\cos \left( {x + \pi /4} \right)}}\) isJEE Mains 2019 Hard
- Let \(z\,\ne -i\) be any complex number such that \(\frac{{z - i}}{{z + i}}\) is a purely imaginary number. Then \(z +\frac {1}{z}\) isJEE Mains 2014 Hard
- If the value of \(\lim _{x \rightarrow 0}(2-\cos x \sqrt{\cos 2 x})^{\left(\frac{x+2}{x^{2}}\right)}\) is equal to \(e^{a}\), then \(a\) is equal to \(.....\)JEE Mains 2021 Hard
- If \(\mathrm{p}\) and \(\mathrm{q}\) are the lengths of the perpendiculars from the origin on the lines, \(x \operatorname{cosec} \alpha-y \sec \alpha=\operatorname{kcot} 2 \alpha\) and \(x \sin \alpha+y \cos \alpha=k \sin 2 \alpha\) respectively, then \(\mathrm{k}^{2}\) is equal to :JEE Mains 2021 Hard
- If the coefficients of \(x^{-2}\) and \(x^{-4}\) in the expansion of \({\left( {{x^{\frac{1}{3}}} + \frac{1}{{2{x^{\frac{1}{3}}}}}} \right)^{18}}\,,\,\left( {x > 0} \right),\) are \(m\) and \(n\) respectively, then \(\frac{m}{n}\) is equal toJEE Mains 2016 Hard
- Let \(A=\sum_{i=1}^{10} \sum_{j=1}^{10} \min \{i, j\}\) and \(B=\sum_{i=1}^{10} \sum_{j=1}^{10}\max \{i, j\}\). Then \(A+B\) is equal toJEE Mains 2022 Hard
More PYQs from JEE Mains
- Let \(f( x)\) be a polynomial of degree four having extreme values at \( x=1 \) and \( x=2\) . If \(\mathop {\lim }\limits_{x \to 0} \left[ {1 + \frac{{f\left( x \right)}}{{{x^2}}}} \right] = 3\),then \(f\left( 2 \right)\) is equal to :JEE Mains 2015 Hard
- Let \(A\) be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let \(B\) be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in \(A \cap B\), which are divisible by 3, is :JEE Mains 2026 Hard
- If \(\lambda>0\), let \(\theta\) be the angle between the vectors \(\vec{a}=\hat{i}+\lambda \hat{j}-3 \hat{k}\) and \(\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}\). If the vectors \(\vec{a}+\vec{b}\) and \(\vec{a}-\vec{b}\) are mutually perpendicular, then the value of \((14 \cos \theta)^2\) is equal toJEE Mains 2024 Medium
- Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between themselves exceeds the number of games that the men played with the women by \(66\), then the number of men who participated in the tournament lies in the intervalJEE Mains 2014 Hard
- If a complex number \(z\) statisfies the equation \(x + \sqrt 2 \,\,\left| {z + 1} \right|\,+ \,i\, = \,0,\) then \(\left| z \right|\) is equal toJEE Mains 2013 Hard
- Let the position vectors of three vertices of a triangle be \(4 \vec{p}+\vec{q}-3 \vec{r},-5 \vec{p}+\vec{q}+2 \vec{r}\) and \(2 \overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{q}}+2 \overrightarrow{\mathrm{r}}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{\vec{p}+\vec{q}+\vec{r}}{4}\) and \(\alpha \vec{p}+\beta \vec{q}+\gamma \vec{r}\) respectively, then \(\alpha+2 \beta+5 \gamma\) is equal to :JEE Mains 2025 Hard