MHT CET · Maths · Vector Algebra
If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are lengths of the sides \(\mathrm{BC}, \mathrm{CA}, \mathrm{AB}\) respectively of \(\triangle \mathrm{ABC}\) and \(\mathrm{H}\) is any
point in the plane of \(\Delta \mathrm{ABC}\) such that a \(\overline{A H}+b \overline{B H}+c \overline{C H}=\overline{0}\), then \(\mathrm{H}\) is the
- A Circumcentre of \(\Delta \mathrm{ABC}\)
- B Incentre of \(\Delta \mathrm{ABC}\)
- C Centroid of \(\Delta \mathrm{ABC}\)
- D Orthocentre of \(\Delta \mathrm{ABC}\)
Answer & Solution
Correct Answer
(B) Incentre of \(\Delta \mathrm{ABC}\)
Step-by-step Solution
Detailed explanation
Consider \(\mathrm{H}\) to be origin.
Then position vector of the vertices \(A, B, C\) are \(\bar{a}, \bar{b}, \bar{c}\) respectively.
We have \(\mathrm{a} \overline{\mathrm{AH}}+\mathrm{b} \overline{\mathrm{BH}}+\mathrm{c} \overline{\mathrm{CH}}=\overline{0}\) i.e. \(\quad \mathrm{a} \overline{\mathrm{a}}+\mathrm{b} \overline{\mathrm{b}}+\mathrm{c} \overline{\mathrm{c}}=0\)
i.e. \(\frac{a \bar{a}+b \bar{b}+c \bar{c}}{a+b+c}=0\), which is position vector of incentre.
Hence \(\mathrm{H}\) is incentre of triangle.
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
- The differential equation \(x \frac{\mathrm{dy}}{\mathrm{d} x}=2 \mathrm{y}\) represents ....MHT CET 2025 Easy
- The area of the region bounded by the curve \(y=2 x-x^2\) and \(\mathrm{X}\)-axis isMHT CET 2021 Medium
- The shortest distance between the lines \(\frac{x+1}{3}=\frac{y-2}{2}=\frac{z+1}{2}\) and \(\frac{x-2}{1}=\frac{y-2}{2}=\frac{z+3}{3}\) isMHT CET 2025 Medium
- Let \(\bar{a}=\alpha \hat{i}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}, \overline{\mathrm{b}}=3 \hat{i}-\hat{\mathrm{j}}+\beta \hat{\mathrm{k}}\) and \(\overline{\mathrm{c}}=\hat{i}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}\) where \(\alpha, \beta \in \mathbb{R}\), be three vectors. If the projection of \(\bar{a}\) on \(\bar{c}\) is \(\frac{10}{3}\) and \(\overline{\mathrm{b}} \times \overline{\mathrm{c}}=-6 \hat{i}+10 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}\), then the value of \((\alpha+\beta)\) is equal toMHT CET 2025 Medium
- \(\int\left[\log (1+\cos x)-x \tan \left(\frac{x}{2}\right)\right] d x=\)MHT CET 2020 Medium
- If \(A^{-1}=\left[\begin{array}{cc}2 & -3 \\ -1 & 2\end{array}\right]\) and \(B^{-1}=\left[\begin{array}{cc}1 & 0 \\ -3 & 1\end{array}\right]\), then \((A B)^{-1}=\)MHT CET 2021 Medium
More PYQs from MHT CET
- Identify the reagent used in the following reaction.
Benzoic acid \(\stackrel{\text { Reagent }}{\Delta}\) Benzoyl chloride + Phosphorous oxychloride + Hydrogen chlorideMHT CET 2023 Easy - Two identical bar magnets each of magnetic moment 'M', separated by some distance are kept perpendicular to each other. The magnetic induction at a point at the same distance 'd' from the centre of magnets, is \(\quad\left(\mu_{0}=\right.\) permeability of free space \()\)MHT CET 2020 Easy
- In an angiosperm a female plant having 2n = 24 is crossed with a male plant having 2n = 12. What will be the chromosome number of the endosperm?MHT CET 2016 Easy
- Two gases A and B having same initial state (P, V, n, T). Now gas A is compressed to \(\frac{\mathrm{V}}{8}\) by isothermal process and other gas \(B\) is compressed to \(\frac{V}{8}\) by adiabatic process. The ratio of final pressure of gas A and B is (Both gases are monoatomic, \(\gamma=5 / 3\) )MHT CET 2024 Medium
- The magnetic field intensity H at the centre of a long solenoid having n turns per unit length and carrying a current I, when no material is kept in it is ( \(\mu_0=\) permeability of free space)MHT CET 2025 Easy
- A body weighs ' \(W\) ' newton on the surface of the earth its weight at a height equal to half the radius of the earth, will beMHT CET 2022 Easy