**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.