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The straight lines 3x + 4y =5 and 4x-3y...

The straight lines `3x + 4y =5 and 4x-3y = 15` interrect at a point `A(3,-1)`. On these linepoints B and C are chosen so that `AB = AC`. Find the possible eqns of the line BC pathrough the point `(1, 2)`

Text Solution

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Let `m_(1)` and `m_(2)` be the slopes of the lines `3x+4y=5` and `4x-3y=15`, respectively.
Then, `m_(1)=-(3)/(4)` and `m_(2)=(4)/(3)`
Clearly, `m_(1)m_(2)=-1`. So, lines `AB` and `AC` are at right angle. Thus, the `DeltaABC` is a right angled isosceless triangle.

Hence, the line `BC` through `(1,2)` will make an angle of `45^(@)` with the given lines. So, the possible equations of `BC` are
`(y-2)=(m+-tan45^(@))/(1overset(-)+tan45^(@))(x-1)`
where, `m="slope of"AB=-(3)/(4)`
`implies(y-2)=(-(3)/(4)+-1)/(1overset(-)+((-3)/(4)))(x-1)`
`implies(y-2)=(-3+-4)/(4+-3)(x-1)`
`implies(y-2)=(1)/(7)(x-1)`
and `(y-2)=-7(x-1)`
`impliesx-7y+13=0`
and `7x+y-9=0`
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