Solution:
We have given A(-2,5) , B(1,-3) and C(a,b) form an isosceles triangle then the value of 6a-16b+19 = ?
We know the distance formula between two points
Now
AB = sqrt[(-3 - (5))² + ((1) - (-2))²]
AB = √73
BC = sqrt[(b - (-3))² + ((a) - 1)²] = sqrt[(b + 3))² + (a - 1)²]
AC = sqrt[(b - 5))² + (a + 2)²]
Since triangle is isosceles, and we know isosceles triangle have two sides are equal, so either AB=BC or AB=AC or BC=AC
Let AB = BC
√73 = sqrt[(b + 3))² + (a - 1)²]
73 = (b + 3))² + (a - 1)² ----------(i)
and AB = AC
√73 = sqrt[(b - 5))² + (a + 2)²]
73 = (b - 5))² + (a + 2)² -------------(ii)
From equation (i) and (ii), we get
=> (b + 3))² + (a - 1)² = (b - 5))² + (a + 2)²
Expanding the equation we get
6a - 16b + 19 = 0 Ans.
