The line `(x+6)/5=(y+10)/3=(z+14)/8` is the hypotenuse of an isosceles right-angled triangle whose opposite vertex is `(7,2,4)dot` Then which of the following in not the side of the triangle? a. `(x-7)/2=(y-2)/(-3)=(z-4)/6` b. `(x-7)/3=(y-2)/6=(z-4)/2` c. `(x-7)/3=(y-2)/5=(z-4)/(-1)` d. none of these

Options

A. `(x-7)/(2)=(y-2)/(-3)=(z-4)/(6)`
B. `(x-7)/(3)=(y-2)/(6)=(z-4)/(2)`
C. `(x-7)/(3)=(y-2)/(5)=(z-4)/(-1)`
D. none of these

Answer

c

Explanation

Given one vertex A(7,2,4) and line
`(x+6)/(5)=(y+10)/(3)=(z+14)/(8)`
General point on above line `B-=(5lamda-6,3lamda-10,8lamda-14)`
Direction ratios of line AB are `lt5lamda-13,3lamda-12,8lamda-18gt`
Direction ratios of line BC are `lt5,3,8gt`
Since angle between AB and BC is `pi//4`, we have
`cos""(pi)/(4)=((5lamda-3)+3(3lamda-12)+8(8lamda-18))/(sqrt(5^(2)
+3^(2)+8^(2)).sqrt((5lamda-13)^(2)+(3lamda-12)^(2)+(8lamda-18)^(2)))`
Squaring and solving we have `lamda=3,2`
Hence, equation of lines are `(x-7)/(2)=(y-2)/(-3)=(z-4)/(6)`
and `(x-7)/(3)=(y-2)/(6)=(z-4)/(2)`