JEE Mains · Maths · STD 12 - 11. three dimension geometry
Let the line \(L\) pass through the point \((0,1,2)\), intersect the line \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) and be parallel to the plane \(2 x+y-3 z=4\). Then the distance of the point \(P(1,-9,2)\) from the line \(L\) is
- A \(9\)
- B \(\sqrt{54}\)
- C \(\sqrt{69}\)
- D \(\sqrt{74}\)
Answer & Solution
Correct Answer
(D) \(\sqrt{74}\)
Step-by-step Solution
Detailed explanation
\(\overline{ AB } \cdot \overrightarrow{ n }\) \(\Rightarrow[(1+2 \lambda) \hat{ i }+(1+3 \lambda) \hat{ j }+(1+4 \lambda) \hat{ k }] \cdot(2 \hat{ i }+\hat{ j }-3 \hat{ k })\) \(2+4 \lambda+1+3 \lambda-3-12 \lambda=0\) \(5 \lambda=0 \Rightarrow \lambda=0\)…
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
- Let the co-ordinates of one vertex of \(\triangle ABC\) be \(A (0,2, \alpha)\) and the other two vertices lie on the line \(\frac{x+\alpha}{5}=\frac{y-1}{2}=\frac{z+4}{3}\). For \(\alpha \in Z\), if the area of \(\triangle ABC\) is \(21\) sq. units and the line segment \(BC\) has length \(2 \sqrt{21}\) units, then \(\alpha^2\) is equal to \(...........\).JEE Mains 2023 Medium
- 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:JEE Mains 2026 Hard
- Let \(A = \left\{ {\left( {x,y} \right):{y^2} \le 4x,y - 2x \ge - 4} \right\}\) .The area of the region \(A\) isJEE Mains 2014 Hard
- Let \(\alpha, \beta\) be the roots of the quadratic equation \(x^2+\sqrt{6} x+3=0\). Then \(\frac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}\) is equal toJEE Mains 2023 Hard
- The equation of a common tangent to the parabolas \(y = x ^{2}\) and \(y =-( x -2)^{2}\) is.JEE Mains 2022 Hard
- Let \((\alpha, \beta, \gamma)\) be the foot of perpendicular from the point \((1,2,3)\) on the line \(\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}\). then \(19(\alpha+\beta+\gamma)\) is equal to :JEE Mains 2024 Hard
More PYQs from JEE Mains
- If \(a_r\) is the coefficient of \(x^{10-r}\) in the Binomial expansion of \((1+x)^{10}\), then \(\sum \limits_{r=1}^{10} r^3\left(\frac{a_r}{a_{r-1}}\right)^2\) is equal toJEE Mains 2023 Hard
- lf \(\int {\frac{{\tan \,\,x\,}}{{1 + \,\tan \,x\, + {{\tan }^2}\,x}}dx} \) \( = x - \frac{K}{{\sqrt A }}{\tan ^{ - 1}}\,\left( {\frac{{K\,\,\tan \,x + 1}}{{\sqrt A }}} \right) + C,\) (\(C\) is a constant ofintegration), then the ordered pair \((K, A)\) is euqal toJEE Mains 2018 Hard
- Let \(\lambda_1, \lambda_2\) be the values of \(\lambda\) for which the points \(\left(\frac{5}{2}, 1, \lambda\right)\) and \((-2,0,1)\) are at equal distance from the plane \(2 x+3 y-6 z+7=0\). if \(\lambda_1 > \lambda_2\), then the distance of the point \(\left(\lambda_1-\lambda_2, \lambda_2, \lambda_1\right)\) from the line \(\frac{x-5}{1}=\frac{y-1}{2}=\frac{z+7}{2}\) is \(............\).JEE Mains 2023 Hard
- The mean and variance of \(8\) observations are \(10\) and \(13.5,\) respectively. If \(6\) of these observations are \(5,7,10,12,14,15,\) then the absolute difference of the remaining two observations isJEE Mains 2020 Hard
- If the system of linear equations \(x_1 + 2x_2 + 3x_3 = 6\) ; \(x_1 + 3x_2 + 5x_3 = 9\) ; \(2x_1 + 5x_2 + ax_3 = b\) is consistent and has infinite number of solutions, thenJEE Mains 2013 Hard
- Let \([\bullet]\) denote the greatest integer function, and let \(f(x)=\min \left\{\sqrt{2} x, x^2\right\}\). Let \(S=\{x \in(-2,2):\) the function \(g ( x )=| x |\left[ x ^2\right]\) is discontinuous at x \(\}\). Then \(\sum_{x \in S} f(x)\) equals :JEE Mains 2026 Easy