If $$\overrightarrow{b}$$ is a vector whose initial point divides the join of $$5\widehat{i}$$ and $$5\widehat{j}$$ in the ratio $$k : 1$$ and whose terminal point is the origin and $$|\vec b| \leq \sqrt{37}$$, then $$k$$ lies in the interval
The point that divides 5$$\widehat{i}$$ and 5$$\widehat{j}$$ in the ratio of $$k : 1$$ is
$$\displaystyle \dfrac{(5 \widehat{j}) k + (5\widehat{i})1}{k + 1}$$
$$\therefore \displaystyle \vec b = \frac{5 \widehat{i} + 5 k \widehat{j}}{k + 1}$$
Also, $$|\vec b| \leq \sqrt{37}$$
$$\Rightarrow \displaystyle \dfrac{1}{k+1} \sqrt{25 + 25 k^2} \leq \sqrt{37}$$
or $$5\sqrt{1 + k^2} \leq \sqrt{37} (k + 1)$$
Squaring both sides, we get
$$25 (1 + k^2) \leq 37 (k^2 + 2k +1)$$
or $$6k^2 + 37k + 6 \geq 0$$ or $$(6k + 1) (k + 6) \geq 0$$
$$k \in (-\infty, - 6] \cup \left [ - \dfrac{1}{6}, \infty \right )$$
Let $$\mathrm{A}\mathrm{B}\mathrm{C}$$ be a triangle and let $$\mathrm{S}$$ be its circumcentre and $$\mathrm{O}$$ be its orthocentre. The $$\overline{\mathrm{S}\mathrm{A}}+\overline{\mathrm{S}\mathrm{B}}+\overline{\mathrm{S}\mathrm{C}}= $$
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Taking $$O$$' as origin and the position vectors of $$A, B$$ are $$\vec i+3\vec{j}-2\vec k, 3\vec{i}+\vec{j}-2\vec{k}$$. The vector $$\overrightarrow{OC}$$ is bisecting the angle $$AOB$$ and if $$C$$ is a point on line $$\overrightarrow{AB}$$ then $$C$$ is
$$ABCD$$ is a quadrilateral, $$E$$ is the point of intersection of the line joining the midpoints of the opposite sides. If $$O$$ is any point and $$\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} = \vec{x OE},$$ then $$x$$ is equal to
Let $$ABC$$ be a triangle whose circumcentre is at P. If the position vectors of $$A, B, C$$ and P are $$\vec {a}, \vec {b}, \vec {c}$$ and $$\dfrac {\vec {a} + \vec {b} + \vec {c}}{4}$$ respectively, then the position vector of the orthocentre of this triangle, is:
Find the position vector of a point $$R$$ which divides the line joining two points $$P$$ and $$Q$$ whose position vectors are $$\hat {i}+2\hat {j}-\hat {k}$$ and $$-\hat {i}+\hat {j}+\hat {k}$$ respectively, in the ratio $$2:1$$, (i) internally (ii) externally
Find the position vector of the mid point of the vector joining the points $$P (2, 3, 4)$$ and $$Q (4, 1, 2)$$.
In a quadrilateral $$PQRS,\ \vec{PQ}=\vec{a}, \vec{QR}=\vec{b}, \vec{SP}=\vec{a} - \vec{b}.\ M$$ is the mid-point of $$QR$$ and $$X$$ is a point on $$SM$$ such that $$\vec{SX}=\dfrac{4}{5}\vec{SM}$$, then $$\vec{PX}$$ is
$$ABCD$$ is a quadrilateral, $$E$$ is the point of intersection of the line joining the midpoints of the opposite sides. If $$O$$ is any point and $$\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} = \vec{x OE},$$ then $$x$$ is equal to
In triangle $$ABC$$, $$\angle A = 30^o$$, $$H$$ is the orthocentre and $$D$$ is the midpoint of $$BC$$. Segment $$HD$$ is produced to $$T$$ such that $$HD = DT$$. The length $$AT$$ is equal to