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Question

Let AB be a line segment of length 4 units with A on line y=2x and B on the line y=x, the locus of middle points of all such line segments is a

A
Parabola
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B
Ellipse
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C
pair of straight lines
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D
circle
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Solution

The correct option is B Ellipse
As given that A lies on line y=2x and B on the line y=x
Let A(t,2t) and B(k,k); AB=4
(tk)2+(2tk)2=4
(tk)2+(2tk)2=16 (1)

and let mid point of AB be (x,y)
t+k2=x,k+2t2=y
t=2(yx),k=4x2y
By equation (1)
(4y6x)2+(6y8x)2=16
(2y3x)2+(3y4x)2=4
25x2+13y236xy4=0
=∣ ∣ahghbfgfc∣ ∣
=∣ ∣2518018130004∣ ∣
0
and
h2<ab So, it's an ellipse

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