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in Co-ordinate geometry by (52.7k points)

ABCD is a variable rectangle having its sides parallel to fixed directions. The vertices B and D lie on x = a and x = a and A lies on the line y = 0. Find the locus of point C.

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See Fig. Let A be (x1, 0), B be (a, y2) and D be (−a, y3). We are given that AB and AD have fixed directions and hence their slopes are constant, say, m1 and m2. Therefore,

y2/a - x1 = m1 and y3/-a - x1 = m2

Further, m1m2 = −1 since ABCD is rectangle.

y2/a - x1 = m1 and y3/a + x1 = 1/m1  .....(1)

Let the coordinates of C be (α ,β). Now, Midpoint of BD ≡ Midpoint of AC

This implies that

By Eqs. (1) and (2), we have

-(m12 - 1)α + m1β =(m12 + 1)a

Therefore, the locus of point C is

m1y = (m2+ 1)a + (m21 - 1)x

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