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If the vectors \(\rm \rm (- \hat i-2 x \hat{j}-3 y \hat{k}) \ and \ (\rm \hat i-3 x \rm\hat j-2 y \rm\hat{k})\)orthogonal to each other, then what is the locus of the point (x, y)? 1. a straight line2. An ellipse3. A circle 4. A parabola |
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Answer» Correct Answer - Option 3 : A circle Concept: If vectors a and b are orthogonal, then \(\rm \vec a.\vec b\) = 0 Equation of circle: x2 + y2 = r2, where r = radius
Calculation: \(\begin{aligned} \rm &\rm (- \hat i-2 x \hat{j}-3 y \hat{k})(\rm \hat i-3 x \rm\hat j-2 y \rm\hat{k})=0\\ &\Rightarrow(-1)(1)+(-2\rm x)(-3\rm x)+(-3\rm y)(-2 \rm y)=0\\ &\Rightarrow 6 \rm x^{2}+6\rm y^{2}=1\\ &\Rightarrow \rm x^{2}+y^{2}=(1 / \sqrt{6})^{2}\\ &\text { Locus of }\rm(x, y) \text { is a circle } \end{aligned}\) Hence, option (3) is correct. |
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