MHT CET · Maths · Three Dimensional Geometry
Let \(M\) and \(N\) be foots of the perpendiculars drawn from the point \(\mathrm{P}(a, a, a)\) on the lines \(x-\mathrm{y}=0, \mathrm{z}=1\) and \(x+\mathrm{y}=0, \mathrm{z}=-1\) respectively and if \(\angle \mathrm{MPN}=90^{\circ}\) then \(a^2=\)
- A 1
- B 4
- C 6
- D 9
Answer & Solution
Correct Answer
(A) 1
Step-by-step Solution
Detailed explanation
\(\mathrm{M}\) is the foot of the perpendicular from \(\mathrm{P}(a, a, a)\) on \(x-y=0, z=1\). Line is \((\lambda, \lambda, 1)\). \(\vec{PM} = (\lambda-a, \lambda-a, 1-a)\). Direction vector \((1,1,0)\). \(\vec{PM} \cdot (1,1,0) = 0 \Rightarrow (\lambda-a) \cdot 1 + (\lambda-a) \cdot 1 + (1-a) \cdot 0 = 0 \Rightarrow 2(\lambda-a)=0 \Rightarrow \lambda=a\). So, \(\mathrm{M}(a, a, 1)\).
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
- \( \int_{0}^{\frac{\pi}{2}} \frac{d x}{1+\cos x}= \)MHT CET 2020 Medium
- Let \(X\) denote the number of hours you study on a Sunday. It is known that.
\(\mathrm{P}(\mathrm{X}=\mathrm{x})=\left\{\begin{array}{cl}
0 \cdot 1 & , \text { if } x=0 \\
\mathrm{k} x & , \text { if } x=1 \text { or } 2 \\
\mathrm{k}(5-x) & , \text { if } x=3 \text { or } 4 \\
0 & , \text { otherwise }
\end{array}\right.\)
where k is constant. Then the probability that you study at least two hours on a Sunday isMHT CET 2025 Medium - If \(y=x \tan y\), then \(\frac{d y}{d x}=\)MHT CET 2021 Easy
- 21 friends were invited for a party. Two round tables can accommodate 12 and 9 friends each. The number of ways of the seating arrangements of friends is .....MHT CET 2025 Medium
- If the population grows at the rate of \(8 \%\) per year, then the time taken for the population to be doubled is (Given \(\log 2=0.6912\) )MHT CET 2021 Easy
- The vectors \(\overline{\mathrm{p}}=\hat{\mathrm{i}}+a \hat{\mathrm{j}}+a^2 \hat{\mathrm{k}}, \overline{\mathrm{q}}=\hat{\mathrm{i}}+b \hat{\mathrm{j}}+b^2 \hat{\mathrm{k}}\) and \(\overline{\mathrm{r}}=\hat{\mathrm{i}}+c \hat{\mathrm{j}}+c^2 \hat{\mathrm{k}}\) are non-coplanar and
\(\left|\begin{array}{lll}
a & a^2 & 1+a^3 \\
b & b^2 & 1+b^3 \\
c & c^2 & 1+c^3
\end{array}\right|=0\)
then the value of \((a b c)\) isMHT CET 2025 Hard
More PYQs from MHT CET
- Which among the following gas is bubbled through the brine solution during the preparation of sodium carbonate in Solvay's process?MHT CET 2020 Easy
- The frequency of a tuning fork is ' \(\mathrm{n}\) ' \(\mathrm{Hz}\) and velocity of sound in air is ' \(\mathrm{V}\) ' \(\mathrm{m} / \mathrm{s}\). When the tuning fork complete ' \(\mathrm{x}\) ' vibrations, the distance traveled by the wave isMHT CET 2021 Medium
- The half life of a first order reaction is \(6.0\) hour. How long will it take for the concentration of reactant to decrease from \(0.4 \mathrm{M}\) to \(0 \cdot 12 \mathrm{M} ?\)MHT CET 2020 Medium
- Which of the following molecules contains maximum number of electrons in antibonding molecular orbitals?MHT CET 2025 Medium
- In communication system, what is used to extend the range of a communication?MHT CET 2020 Easy
- An observer on sea-coast counts 45 waves in one minute. If the wavelength of the waves is 7 m , then the velocity of the waves will beMHT CET 2025 Easy