question_answer

If the extremities of a line segment of lengths \[\ell \], moves in two fixed perpendicular straight lines, then the locus of that point which divides this line segment in the ratio 1 : 2, is-

A) A parabola        

B)        An ellipse

C) A hyperbola      

D)        None of these

Correct Answer: B

Solution :

[b] Let the two fixed perpendicular straight lines be the co-ordinate axis let P(h, k) be the point whose locus is required Let \[PA:PB=1:2\] Then \[PA=\frac{\ell }{3}\] and \[PB=\frac{2\ell }{3}\] \[k=\frac{\ell }{3}sin\theta \]     or \[3k=\ell \,sin\theta \]                 ?.... (i) and   \[h=\frac{2\ell }{3}\cos \theta \] or \[\frac{3h}{2}=\ell \cos \theta \] .....(ii) Squaring and adding (i) and (ii) we get \[9{{k}^{2}}+\frac{9{{h}^{2}}}{4}={{\ell }^{2}}\] \[\therefore \text{ }locus~~~9{{x}^{2}}+36{{y}^{2}}=4{{\ell }^{2}}\] which is an ellipse