JEE Mains · Maths · STD 12 - 11. three dimension geometry
Let \(P(\alpha,\beta,\gamma)\) be the point on the line \(\frac{x-1}{2}=\frac{y+1}{-3}=z\) at a distance \(4\sqrt{14}\) from the point (1, -1, 0) and nearer to the origin. Then the shortest distance, between the lines \(\frac{x-\alpha}{1}=\frac{y-\beta}{2}=\frac{z-\gamma}{3}\) and \(\frac{x+5}{2}=\frac{y-10}{1}=\frac{z-3}{1}\), is equal to
- A \(7\sqrt{\frac{5}{4}}\)
- B \(4\sqrt{\frac{7}{5}}\)
- C \(4\sqrt{\frac{5}{7}}\)
- D \(2\sqrt{\frac{7}{4}}\)
Answer & Solution
Correct Answer
(B) \(4\sqrt{\frac{7}{5}}\)
Step-by-step Solution
Detailed explanation
Let \(P (2 \lambda+1,-3 \lambda-1, \lambda)\) Then \(4 \lambda^2+9 \lambda^2+\lambda^2=16 \cdot 14 \Rightarrow \lambda= \pm 4 \Rightarrow-4\) (nearer to origin) \(\therefore P (-7,11,-4)\) ∴ Shortest distance…
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
- If the line, \(\frac{{x\, - \,1}}{2}\, = \,\frac{{y\, + \,1}}{3}\, = \,\frac{{z\, - \,2}}{4}\) meets the plane, \(x + 2y + 3z = 15\) at a point \(P,\) then the distance of \(P\) from the origin isJEE Mains 2019 Medium
- Let \(\mathrm{z}=\frac{1-i \sqrt{3}}{2}, i=\sqrt{-1} .\) Then the value of \(21+\left(z+\frac{1}{z}\right)^{3}+\left(z^{2}+\frac{1}{z^{2}}\right)^{3}+\left(z^{3}+\frac{1}{z^{3}}\right)^{3}+\ldots+\left(z^{21}+\frac{1}{z^{21}}\right)^{3}\) is .... .JEE Mains 2021 Hard
- For the two circles \(x^2 + y^2 = 16\) and \(x^2 + y^2 -2y = 0,\) there is/areJEE Mains 2014 Hard
- A group of students comprises of \(5\) boys and \(n\) girls. If the number of ways, in which a team of \(3\) students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is \(1750\), then \(n\) is equal toJEE Mains 2019 Hard
- If the fourth term in the binomial expansion of \(\left(\sqrt{\frac{1}{x^{1+\log _{10} x}}}+x^{\frac{1}{12}}\right)^{6}\) is equal to \(200\), and \(x > 1\), then the value of \(x\) isJEE Mains 2019 Hard
- Let \(\alpha, \beta\) be the roots of the equation \(x^2 - 3x + r = 0\), and \(\dfrac{\alpha}{2}, 2\beta\) be the roots of the equation \(x^2 + 3x + r = 0\). If the roots of the equation \(x^2 + 6x = m\) are \(2\alpha + \beta + 2r\) and \(\alpha - 2\beta - \dfrac{r}{2}\), then \(m\) is equal to:JEE Mains 2026 Hard
More PYQs from JEE Mains
- For each \(x\,\in R,\) let \([x]\) be the greatest integer less than or equal to \(x.\) Then \(\mathop {\lim }\limits_{x \to {0^ + }} \frac{{x([x] + [x])\,\sin \,[x]}}{{\left| x \right|}}\) is equal toJEE Mains 2019 Hard
- If the distance between planes , \(4x - 2y-4z + 1 = 0\) and \(4x -2y-4z+ d = 0\) is \(7,\) then \(d\) isJEE Mains 2014 Medium
- Let \(R\) be the focus of the parabola \(y^2=20 x\) and the line \(y=m x+c\) intersect the parabola at two points \(P\) and \(Q\). Let the point \(G(10,10)\) be the centroid of the triangle \(P Q R\). If \(c-m=6\), then \(( PQ )^2\) isJEE Mains 2023 Hard
- Let \(y=y(x)\) be the solution of the differential equation \(e^{x} \sqrt{1-y^{2}} d x+\left(\frac{y}{x}\right) d y=0, y(1)=-1\) Then the value of \((y(3))^{2}\) is equal to:JEE Mains 2021 Hard
- If vertex of a parabola is \((2,-1)\) and the equation of its directrix is \(4 x-3 y=21\), then the length of its latus rectum isJEE Mains 2022 Easy
- Let for the \(9^{\text {th }}\) term in the binomial expansion of \((3+6 x)^{n}\), in the increasing powers of \(6 x\), to be the greatest for \(x=\frac{3}{2}\), the least value of \(n\) is \(n_{0}\). If \(k\) is the ratio of the coefficient of \(x ^{6}\) to the coefficient of \(x ^{3}\), then \(k + n _{0}\) is equal to.JEE Mains 2022 Hard