JEE Mains · Maths · STD 12 - 11. three dimension geometry
Let the line \(\mathrm{L}\) be the projection of the line \(\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-4}{2}\) in the plane \(x-2 y-z=3 .\) If \(d\) is the distance of the point \((0,0,6)\) from \(\mathrm{L}\), then \(\mathrm{d}^{2}\) is equal to .... .
- A \(48\)
- B \(26\)
- C \(14\)
- D \(1\)
Answer & Solution
Correct Answer
(B) \(26\)
Step-by-step Solution
Detailed explanation
\(\mathrm{L}_{1}: \frac{x-1}{2}=\frac{y-3}{1}=\frac{z-4}{2}\) for foot of \(\perp \mathrm{r}\) of \((1,3,4)\) on \(x-2 y-z-3=0\) \((1+t)-2(3-2 t)-(4-t)-3=0\) \(\Rightarrow \mathrm{t}=2\) So foot of \(\perp \mathrm{r} \triangleq(3,-1,2)\) and point of intersection of…
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 \(\vec{a}=-\hat{i}-\hat{j}+\hat{k}, \vec{a} \cdot \vec{b}=1\) and \(\vec{a} \times \vec{b}=\hat{i}-\hat{j}\). Then \(\vec{a}-6 \vec{b}\) is equal toJEE Mains 2023 Medium
- Let \(f\left( x \right) = {x^2} + \frac{1}{{{x^2}}}\) and \(g\left( x \right) = x - \frac{1}{x},\;x \in R - \left\{ { - 1,1,0} \right\}\). If \(h\left( x \right) = \frac{{f\left( x \right)}}{{g\left( x \right)}}\) then the local minimum value of \(h\left( x \right)\) is:JEE Mains 2018 Hard
- Let \(f :[0,1] \rightarrow R\) be a twice differentiable function in \((0,1)\) such that \(f (0)=3\) and \(f (1)=5\). If the line \(y=2 x+3\) intersects the graph of \(f\) at only two distinct points in \((0,1)\), then the least number of points \(x \in(0,1)\), at which \(f ^{\prime \prime}( x )=0\), is\(......\)JEE Mains 2022 Hard
- The number of solutions of the equation \(1 + {\sin ^4}\,x = {\cos ^2}\,3x,x\,\in \,\left[ { - \frac{{5\pi }}{2},\frac{{5\pi }}{2}} \right]\) isJEE Mains 2019 Hard
- Let \(z\) be a complex number such that \(|z|=1\). If \(\frac{2+\mathrm{k}^2 \mathrm{z}}{\mathrm{k}+\overline{\mathrm{z}}}=\mathrm{kz}, \mathrm{k} \in \mathbf{R}\), then the maximum distance of \(\mathrm{k}+\mathrm{ik}^2\) from the circle \(|\mathrm{z}-(1+2 \mathrm{i})|=1\) is:JEE Mains 2025 Hard
- If \(z = x + iy\) satisfies \(|z|-2=0\) and \(|z-i|-|z+5 i|=0\), thenJEE Mains 2022 Hard
More PYQs from JEE Mains
- The value of \(\int\limits_{0}^{2\pi } {\left[ {\sin \,2x\left( {1 + \cos \,3x} \right)} \right]} \,dx\) where \([t]\) denotes the greatest integer function, isJEE Mains 2019 Hard
- If \(A = \left[ {\begin{array}{*{20}{c}}1&2&2\\2&1&{ - 2}\\a&2&b\end{array}} \right]\) is a matrix satisfying the equation \(AA^T=9I \) where\( I\) is \(3×3\) identity matrix, then the ordered pair \((a, b)\) is equal to:JEE Mains 2015 Medium
- Let \(P \left(\frac{2 \sqrt{3}}{\sqrt{7}}, \frac{6}{\sqrt{7}}\right), Q , R\) and \(S\) be four points on the ellipse \(9 x^2+4 y^2=36\). Let \(P Q\) and \(RS\) be mutually perpendicular and pass through the origin. If \(\frac{1}{( PQ )^2}+\frac{1}{( RS )^2}=\frac{ p }{ q }\), where \(p\) and \(q\) are coprime, then \(p+q\) is equal to \(.........\).JEE Mains 2023 Hard
- The vector equation of the plane through the line of intersection of the planes \(x + y + z = 1\) and \(2x + 3y + 4z = 5\) which is perpendicular to the plane \(x -y + z = 0\) isJEE Mains 2019 Hard
- Let PQ and MN be two straight lines touching the circle \( x^{2}+y^{2}-4x-6y-3=0 \) at the points A and B respectively. Let O be the centre of the circle and \( \angle AOB=\pi/3. \) Then the locus of the point of intersection of the lines PQ and MN is:JEE Mains 2026 Hard
- Let \(l_{1}\) be the line in \(xy\)-plane with \(x\) and \(y\) intercepts \(\frac{1}{8}\) and \(\frac{1}{4 \sqrt{2}}\) respectively, and \(l_{2}\) be the line in \(zx\)-plane with \(x\) and \(z\) intercepts \(-\frac{1}{8}\) and \(-\frac{1}{6 \sqrt{3}}\) respectively. If \(d\) is the shortest distance between the line \(l_{1}\) and \(l_{2}\), then \(d ^{-2}\) is equal toJEE Mains 2022 Hard