A(3,0) and B(6,0) are two fixed points and U(`x_1,y_1`) is a variable point of the plane .AU and BU meets the y axis at C and D respectively and AD meets OU at V. Then for any position of U in the plane CV passes through fixed point (p,q) whose distance from origin is____units

Options

A. 1units
B. 2 units
C. 3 units
D. 4 units

Answer

B

Explanation

Equation of AU is
`y - y_(1) = (0-y_(1))/(3-x_(1)) (x-x_(1))`
So that the coordinates of C are `(0,3y_(1)//(3-x_(1)))`
Similarly, the coordinates of D are `(0,6y_(1)//(6-x_(1)))`
Now, equation of AD is `(x)/(3) +(y(6-x_(1)))/(6y_(1)) =1` (i)
and equation of OU is `y_(1)x = x_(1)y` (ii)

Solving (i) and (ii), we get
`(x_(1)y)/(3y_(1)) +(y(6-x_(1)))/(6y_(1)) =1`
`rArr y(2x_(1)+6-x_(1)) = 6y_(1)`
`rArr y =(6y_(1))/(6+x_(1)) rArr x =(6x_(1))/(6+x_(1))`
Hence, the coordinates of V are `((6x_(1))/(6+x_(1)),(6y_(1))/(6+x_(1)))`
Therefore, equation of CV is `y -(3y_(1))/(3-x_(1)) = ((6y_(1))/(6+x_(1))-(3y_(1))/(3-x_(1)))/((6x_(1))/(6+x_(1)) -0) (x-0)`
`rArr y =(3y_(1))/(3-x_(1)) -(9x_(1)y_(1))/(6x_(1)(3-x_(1)))x rArr y=(3y_(1))/(3-x_(1))(1-(x)/(2))`
which passes through the point (2,0) for all values of `(x_(1),y_(1))`