# A variable straight line passes through the point P(α,β)straight line passes through the points A and B, respectively.

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A variable straight line passes through the point P(α,β)straight line passes through the points A and B, respectively. If the parallelogram OACB is completed, then the locus of the vertex C (O is the origin) is

(a)   x/ β + y/α = 1

(b)   β/x + α/y = 1

(c)  α/x + β/y = 1

(d)  αx + βy = (α + β)xy

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Correct option  (c)  α/x + β/y = 1

Explanation :

See Fig .

Let the line AB be

x/a + y/b = 1

where A = (a, 0) and B = (0, b). This line passes through P (α,β) which implies that

α/a + β/b = 1  ...(1)

Let C(h, k) be the fourth vertex. Therefore, h = a and k = b. Hence, from Eq. (1), we have

α/h + β/k = 1

Therefore, the locus of C is

α/x + β/y = 1