Find the Equations of the straight line with the help of the given information.

Question

Suppose two straight lines 3x+4y=5 & 4x-3y=15 cut each other at the point A. Take 2 points B & C on those 2 lines, such that AB=AC. If line BC passes through the point (1,2), then find the possible equations of the straight line BC.

The given equations are perpendicular to each other & given that AB=AC. But I am unable to use the given information to find the solution.

Please help

Thank you

Answer 1

I guess you mean AB=AC instead AB=BC

1) It is not difficult to check that line ax+by+c=0 is ortogonal to vector (a,b).

2) Target lines are parallel to one of the two bisectrix of the given lines intersection angle.

3) Since we have 3x+4y=5 & 4x-3y=15 equations, then n₁=(3,4) and n₂=(4,-3) - ortogonal vectors with same length 5. So one of the bisectrix vector equal n1+n2 = (7,1), second is ortogonal to this (-1,7).

4) So answer is two family of lines: 7x+y+c=0 and -x+7y+c=0

Answer 2

Well, $ABC$ is an isosceles triangle since $AB=BC$. Still it also has a right angle in $A$. This is impossible... There is no such points $B$ and $C$ such that $AB = BC$.