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Question 35: Two points on a ground are 1 m apart. If a cow moves in the field in such a way that it's distance from the two points is always in ratio 3: 2 then

- the cow moves in a straight line
- the cow moves in a circle
- the cow moves in a parabola
- the cow moves in a hyperbola

the cow moves in a circle

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Let us say that the cow moves between the points O and A which are 1m apart.

For convenience, Let us fix O at (0,0) and A at (1,0).

Let us assume that B is at some point (x,y) where the cow is grazing. Let P be the perpendicular dropped from B on OA.

So OP = x and PA = 1-x

OB^{2}= OP^{2}+ BP^{2}= x^{2}+ y^{2}

OB =\\sqrt{x^{2} + y^{2}}\\)

BA^{2}= AP^{2}+ BP^{2}

BA^{2}= (1-x)^{2}+ y^{2}

BA = \\sqrt{(1-x)^{2} + y^{2}}\\)

Also, \\frac{OB}{BA}\\) = \\frac{3}{2}\\)

\\frac{\sqrt{x^{2} + y^{2}}}{\sqrt{(1-x)^{2} + y^{2}}}\\) = \\frac{3}{2}\\)

\\frac{x^{2} + y^{2}}{(1-x)^{2} + y^{2}}\\) = \\frac{9}{4}\\)

4(x^{2}+ y^{2}) = 9((1−x)^{2}+ y^{2}) = 9 (1 + x^{2}– 2x + y^{2})

4 x^{2}+ 4 y^{2}= 9 + 9 x^{2}– 18x + 9y^{2}

5x2 + 5y2 – 18x +9 = 0, This is an equation of locus of a Circle.

Hence, the cow moves in a Circular path.

The question is**"Two points on a ground are 1 m apart. If a cow moves in the field in such a way that it's distance from the two points is always in ratio 3: 2 then"**

Choice B is the correct answer

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