A) Circle
B) Ellipse
C) Straight line
D) Hyperbola
Correct Answer: B
Solution :
Let fixed direction be \[OA\] and \[OB\]inclined at a constant angle \[\alpha \]and\[AB=c,\] |
Let \[\angle BAO=\theta \,\,BC=c\,\sin \theta \]and \[AC=c\,\cos \theta .\] |
\[\therefore \,OC=c\sin \theta .\cot \alpha \] |
Equation of line passing through A and perpendicular to \[OB\]is |
\[y=-\cot \alpha (x-c\sin \theta \cot \alpha -\cos \theta )\] |
And equation of \[BC\]is \[x=c\,\sin \theta .\cot \alpha \] |
\[\therefore \]Orthocenter is \[(c\sin \theta .\cot \alpha ,c\,\cos \theta .\cot \alpha )\] |
Eliminating \[\theta \] from \[x=c\sin \theta \cot \alpha \]and |
\[y=c\cos \theta \cot \alpha \] |
\[\Rightarrow \]Required locus is \[{{x}^{2}}+{{y}^{2}}={{c}^{2}}{{\cot }^{2}}\alpha ,\]which the equation of a circle is. |
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