A bar of given length moves with its extremities on two fixed straight lines at right angles. Show that any point on the bar describes an ellipse.

Options

A. circle
B. parabola
C. ellipse
D. hyperbola

Answer

C

Explanation

Let the fixed straight lines be along the coordinates axes and AB the bar of length /such that its extremities A (a, 0 ) and B(0, b) are on the coordinates axes. Then,
`a^(2)+b^(2)=l^(2)" "...(i)`
Let P (h,k) be a point marked on the bar such that it divides the bar AB in the ratio `lambda : 1`. Then,
`h=(a)/(lambda+1) and k = (blambda)/(lambda+1)rArr a =(lambda+1)and b = ((lambda+1)k)/(lambda)`
Substituting these values in (i), we get
`(lambda+1)^(2)h^(2)+((lambda+1)^(2)k^(2))/(lambda^(2))=l^(2)`
Hence, the locus of P(h, k) is
`(lambda+1)^(2)x^(2)+(lambda+1)^(2)/(lambda^(2))y^(2)=l^(2)`
ro, `(x^(2))/(((l)/(lambda+1))^(2))+(y^(2))/(((lambdal)/(lambda+1))^(2))=1` which represents an ellipse.