Question:
Let O\((0,0)\), A\((-1,2)\) and B\((1,3)\) be the vertices of \(△\)OAB. The bisector of angle O intersects side AB at point D. The value of \(\overset{⃗}{OD}\cdot \overset{⃗}{AB}\) is equal to...
Let O\((0,0)\), A\((-1,2)\) and B\((1,3)\) be the vertices of \(△\)OAB. The bisector of angle O intersects side AB at point D. The value of \(\overset{⃗}{OD}\cdot \overset{⃗}{AB}\) is equal to...
Show Hint
The angle bisector divides the opposite side in the ratio of the adjacent sides.
Updated On: Oct 1, 2026
- \(-5(\sqrt{2}+1)\)
- \(5(1-\sqrt{2})\)
- \(5(\sqrt{2}+1)\)
- \(5(\sqrt{2}-1)\)
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Understand the concept
The bisector of angle \(O\) meets \(AB\) at \(D\) so that \(\dfrac{AD}{DB} = \dfrac{OA}{OB}\) (angle bisector theorem).
Step 2: Find the lengths
\(OA = \sqrt{1 + 4} = \sqrt{5}\) and \(OB = \sqrt{1 + 9} = \sqrt{10}\). So \(AD : DB = \sqrt{5} : \sqrt{10} = 1 : \sqrt{2}\).
Step 3: Find D
By the section formula, \(\overrightarrow{OD} = \dfrac{\sqrt{2}\,\overrightarrow{OA} + 1\cdot\overrightarrow{OB}}{\sqrt{2} + 1} = \dfrac{(-\sqrt{2} + 1,\ 2\sqrt{2} + 3)}{\sqrt{2} + 1}\).
Step 4: Dot product
\(\overrightarrow{AB} = (2, 1)\). Then \(\overrightarrow{OD}\cdot\overrightarrow{AB} = \dfrac{2(1 - \sqrt{2}) + 2\sqrt{2} + 3}{\sqrt{2} + 1} = \dfrac{5}{\sqrt{2} + 1} = 5(\sqrt{2} - 1)\), option (D).
Final Answer:
The dot product is 5(sqrt 2 - 1). This is option (D). \[ \boxed{\text{(D) }5(\sqrt{2}-1)} \]
The bisector of angle \(O\) meets \(AB\) at \(D\) so that \(\dfrac{AD}{DB} = \dfrac{OA}{OB}\) (angle bisector theorem).
Step 2: Find the lengths
\(OA = \sqrt{1 + 4} = \sqrt{5}\) and \(OB = \sqrt{1 + 9} = \sqrt{10}\). So \(AD : DB = \sqrt{5} : \sqrt{10} = 1 : \sqrt{2}\).
Step 3: Find D
By the section formula, \(\overrightarrow{OD} = \dfrac{\sqrt{2}\,\overrightarrow{OA} + 1\cdot\overrightarrow{OB}}{\sqrt{2} + 1} = \dfrac{(-\sqrt{2} + 1,\ 2\sqrt{2} + 3)}{\sqrt{2} + 1}\).
Step 4: Dot product
\(\overrightarrow{AB} = (2, 1)\). Then \(\overrightarrow{OD}\cdot\overrightarrow{AB} = \dfrac{2(1 - \sqrt{2}) + 2\sqrt{2} + 3}{\sqrt{2} + 1} = \dfrac{5}{\sqrt{2} + 1} = 5(\sqrt{2} - 1)\), option (D).
Final Answer:
The dot product is 5(sqrt 2 - 1). This is option (D). \[ \boxed{\text{(D) }5(\sqrt{2}-1)} \]
Was this answer helpful?
0
0
Top MHT CET Mathematics Questions
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
- If $\int\limits^{K}_0 \frac{dx}{2 + 18 x^2} = \frac{\pi}{24}$, then the value of K is
- If $\int^{\pi/2}_{0} \log\cos x dx =\frac{\pi}{2} \log\left(\frac{1}{2}\right)$ then $ \int^{\pi/2}_{0} \log\sec x dx = $
- The point on the curve $y = \sqrt{x - 1}$ where the tangent is perpendicular to the line $2x + y - 5 = 0 $ is
View More Questions
Top MHT CET Vectors Questions
- If \(\bar{a}, \bar{b}, \bar{c}\) are three unit vectors such that \(|\bar{a} + \bar{b}|^2 + |\bar{a} + \bar{c}|^2 = 8\), then \(|\bar{a} + 3\bar{b}|^2 + |\bar{a} + 3\bar{c}|^2 =\)
- If $\bar{a}$ and $\bar{b}$ are unit vectors and $\theta$ is the angle between them, then $\bar{a} + \bar{b}$ is a unit vector when $\theta$ is
- If $\tan 3\theta = \cot \theta$, then $\theta =$
- The sum of the degree and order of the differential equation $\sqrt{\frac{d^2y}{dx^2}} = \sqrt[5]{\frac{dy}{dx}} - 5$ is
- In a game, 3 coins are tossed. A person is paid Rs \( 150 \) if he gets all heads or all tails and he is supposed to pay ₹\( 50 \) if he gets one head or two heads. The amount he can expect to win lose on an average per game in ₹ is
View More Questions
Top MHT CET Questions
- During r- DNA technology, which one of the following enzymes is used for cleaving DNA molecule ?
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- The temperature of $32^{\circ}C$ is equivalent to
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- The heat of formation of water is $ 260\, kJ $ . How much $ H_2O $ is decomposed by $ 130\, kJ $ of heat ?
View More Questions