JEE Mains · Maths · STD 11 - 11. introduction to three dimensional geometry
Let the point A be the foot of perpendicular drawn from the point \(P(a, b, 0)\) on the line \(\dfrac{x-1}{2} = \dfrac{y-2}{1} = \dfrac{z-\alpha}{3}\). If the midpoint of the line segment PA is \(\left(0, \dfrac{3}{4}, \dfrac{-1}{4}\right)\), then the value of \(a^2 + b^2 + \alpha^2\) is equal to:
- A \(1\)
- B \(2\)
- C \(6\)
- D \(9\)
Answer & Solution
Correct Answer
(A) \(1\)
Step-by-step Solution
Detailed explanation
Let \(P = (a, b, 0)\) and the midpoint of \(PA\) be \(M = \left(0, \dfrac{3}{4}, -\dfrac{1}{4}\right)\). Using the midpoint formula, the coordinates of \(A\) are given by \(2M - P\): \(A = \left(-a, \dfrac{3}{2} - b, -\dfrac{1}{2}\right)\) Since \(A\) lies on the given line…
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 \(\alpha \in(0, \infty)\) and \(A=\left[\begin{array}{lll}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]\). If \(\operatorname{det}\left(\operatorname{adj}\left(2 \mathrm{~A}-\mathrm{A}^{\mathrm{T}}\right) \cdot \operatorname{adj}\left(\mathrm{A}-2 \mathrm{~A}^{\mathrm{T}}\right)\right)=2^8\), then \((\operatorname{det}(\mathrm{A}))^2\) is equal to :JEE Mains 2024 Hard
- A normal to the hyperbola, \(4x^2 - 9y^2\, = 36\) meets the co-ordinate axes \(x\) and \(y\) at \(A\) and \(B\), respectively . If the parallelogram \(OABP\) ( \(O\) being the origin) is formed, then the locus of \(P\) isJEE Mains 2018 Hard
- Let \([\mathrm{x}]\) denote the greatest integer less than or equal to \(\mathrm{x}\). Then, the values of \(x \in R\) satisfying the equation \(\left[e^{x}\right]^{2}+\left[e^{x}+1\right]-3=0\) lie in the interval:JEE Mains 2021 Hard
- Two poles, \(\mathrm{AB}\) of length \(a\) metres and \(\mathrm{CD}\) of length \(\mathrm{a}+\mathrm{b}(\mathrm{b} \neq \mathrm{a})\) metres are erected at the same horizontal level with bases at \(\mathrm{B}\) and \(\mathrm{D} .\) If \(\mathrm{BD}=\mathrm{x}\) and \(\tan \angle\,ACB=\frac{1}{2}\), then:JEE Mains 2021 Hard
- An are \(P Q\) of a circle subtends a right angle at its centre \(O\). The mid point of the arc \(P Q\) is \(R\). If \(\overline{O P}=\vec{u}, \overline{O R}=\vec{v}\) and \(\overrightarrow{O Q}=\alpha \vec{u}+\beta \vec{v}\), then \(\alpha, \beta^2\) are the roots of the equationJEE Mains 2023 Hard
- The number of terms in an \(A .P.\) is even ; the sum of the odd terms in it is \(24\) and that the even terms is \(30\). If the last term exceeds the first term by \(10\frac{1}{2}\) , then the number of terms in the \(A.P.\) isJEE Mains 2014 Hard
More PYQs from JEE Mains
- Let \(z_1, z_2\) and \(z_3\) be three complex numbers on the circle \(|z|=1\) with \(\arg \left(z_1\right)=\frac{-\pi}{4}, \arg \left(z_2\right)=0\) and \(\arg \left(z_3\right)=\frac{\pi}{4}\). If \(\left|z_1 \bar{z}_2+z_2 \bar{z}_3+z_3 \bar{z}_1\right|^2=\alpha+\beta \sqrt{2}, \alpha, \beta \in \mathbf{Z}\), then the value of \(\alpha^2+\beta^2\) is :JEE Mains 2025 Medium
- Let \(S _{1}, S _{2}\) and \(S _{3}\) be three sets defined as \(S _{1}=\{ z \in C :| z -1| \leq \sqrt{2}\}\) ; \(S _{2}=\{ z \in C : \operatorname{Re}((1- i ) z ) \geq 1\}\) ; \(S _{3}=\{ z \in C : \operatorname{Im}( z ) \leq 1\}\) Then the set \(S _{1} \cap S _{2} \cap S _{3}\)JEE Mains 2021 Hard
- In the expansion of \((1+x)\left(1-x^2\right)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5, x \neq 0\), the sum of the coefficient of \(x^3\) and \(x^{-13}\) is equal toJEE Mains 2024 Hard
- Let \(y=y(x)\) be the solution of the differential equation \(x\sqrt{1-x^2}\,dy + \left(y\sqrt{1-x^2} - x\cos^{-1}x\right)dx = 0\), \(x \in (0, 1)\), \(\displaystyle\lim_{x\to 1^-} y(x) = 1\). Then \(y\left(\dfrac{1}{2}\right)\) equals:JEE Mains 2026 Medium
- The plane which bisects the line segment joining the points \((-3, -3, 4)\) and \((3, 7, 6)\) at right angles, passes through which one of the following points?JEE Mains 2019 Hard
- Let \(p , q \in R\) and \((1-\sqrt{3} i )^{200}=2^{199}( p + iq )\), \(i =\sqrt{-1}\) Then \(p + q + q ^2\) and \(p - q + q ^2\) are roots of the equation.JEE Mains 2023 Hard