A stair-case of length l rests against a vertical wall and a floor of a room,. Let P be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio 1 : 2. If the stair-case begins to slide on the floor, then the locus of P is :

  • Option 1)

    an ellipse of eccentricity \frac{1}{2}\;

  • Option 2)

    an ellipse of eccentricity \frac{\sqrt{3}}{2}

  • Option 3)

    a circle of radius \frac{l}{2}

  • Option 4)

    a circle of radius \frac{\sqrt{3}}{2}l

 

Answers (1)

As we learnt in

Selection formula -

x= \frac{mx_{2}+nx_{1}}{m+n}

y= \frac{my_{2}+ny_{1}}{m+n}

- wherein

If P(x,y) divides the line joining A(x1,y1) and B(x2,y2) in ration m:n

 

 and

Eccentricity -

e= \sqrt{1-\frac{b^{2}}{a^{2}}}

- wherein

For the ellipse  

\frac{x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}= 1

 We know length = l

so, x2, + y2 = l2

now using section formula

P(h, k) is (\frac{x}{3},\frac{2y}{3})

so, h=\frac{x}{3}; k=\frac{2y}{3}

x = 3h; y=\frac{3}{2}k

9h^{2}+\frac{9k^{2}}{4}=l^{2}, an ellipse

\frac{h^{2}}{\frac{l^{2}}{9}}+\frac{k^{2}}{\frac{4l^{2}}{9}}=1

soe=\sqrt{\frac{1-l^2}{9\times \frac{4l^2}{9}}} = \frac{\sqrt{3}}{2}


Option 1)

an ellipse of eccentricity \frac{1}{2}\;

this is incorrect

Option 2)

an ellipse of eccentricity \frac{\sqrt{3}}{2}

this is correct

Option 3)

a circle of radius \frac{l}{2}

this is incorrect

Option 4)

a circle of radius \frac{\sqrt{3}}{2}l

this is incorrect

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