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P and Q are two points on a line passing...

P and Q are two points on a line passing through (2, 4) and having slope m. If a line segment AB subtends a right angles at P and Q, where A(0, 0) and B(6,0), then range of values of m is

A

A. `((2-3sqrt(2))/(4),(2+3sqrt(2))/(4))`

B

B. `(-oo,(2-3sqrt(2))/(4))uu((2+3sqrt(2))/(4),oo)`

C

C. `(-4,4)`

D

D. `(-oo,-4)uu(4,oo)`

Text Solution

Verified by Experts

The correct Answer is:
B
AB subtends right angle at P and Q on variable line.

So, AB is a diameter of circle whose chord is a variable line. Equation of circle is:
`x (x-6) +y xx y =0`
or `x^(2) +y^(2) -6x =0` (i)
Equation of line through (2,4) is:
`y-4 =m(x-2)`
or `y = mx +(4-2m)` (ii)
Line (ii) is a chord if `|(3m+(4-2m))/(sqrt(1+m^(2)))| lt3`
`rArr 8m^(2) - 8m - 7 gt 0`
`m in (-oo,(2-3sqrt(2))/(4))uu((2+3sqrt(2))/(4),oo)`
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