1. Find the point on the y-axis which is equidistant from `A(6,5)` and `B(-4,3)`

Solution:
Let `P(0,y)` be the point on Y-axis, which is the equidistance form `A(6,5)` and `B(-4,3)`.

`:. PA=PB`

`=>PA^2=PB^2`

`=>(0-6)^2+(y-5)^2=(0+4)^2+(y-3)^2`

`=>y^2-10y+25+36=y^2-6y+9+16`

`=>-10y+6y=9+16-25-36`

`=>-4y=-36`

`=>y=9`

Thus, the required point is `P(0,9).`

---

2. Find the point on the y-axis which is equidistant from `A(5,-2)` and `B(-3,2)`

Solution:
Let `P(0,y)` be the point on Y-axis, which is the equidistance form `A(5,-2)` and `B(-3,2)`.

`:. PA=PB`

`=>PA^2=PB^2`

`=>(0-5)^2+(y+2)^2=(0+3)^2+(y-2)^2`

`=>y^2+4y+4+25=y^2-4y+4+9`

`=>4y+4y=4+9-4-25`

`=>8y=-16`

`=>y=-2`

Thus, the required point is `P(0,-2).`

---

3. Find the point on the y-axis which is equidistant from `A(-4,3)` and `B(5,2)`

Solution:
Let `P(0,y)` be the point on Y-axis, which is the equidistance form `A(-4,3)` and `B(5,2)`.

`:. PA=PB`

`=>PA^2=PB^2`

`=>(0+4)^2+(y-3)^2=(0-5)^2+(y-2)^2`

`=>y^2-6y+9+16=y^2-4y+4+25`

`=>-6y+4y=4+25-9-16`

`=>-2y=4`

`=>y=-2`

Thus, the required point is `P(0,-2).`

---

4. Find the point on the y-axis which is equidistant from `A(6,-1)` and `B(2,3)`

Solution:
Let `P(0,y)` be the point on Y-axis, which is the equidistance form `A(6,-1)` and `B(2,3)`.

`:. PA=PB`

`=>PA^2=PB^2`

`=>(0-6)^2+(y+1)^2=(0-2)^2+(y-3)^2`

`=>y^2+2y+1+36=y^2-6y+9+4`

`=>2y+6y=9+4-1-36`

`=>8y=-24`

`=>y=-3`

Thus, the required point is `P(0,-3).`

---

5. Find the point on the y-axis which is equidistant from `A(2,3)` and `B(-4,1)`

Solution:
Let `P(0,y)` be the point on Y-axis, which is the equidistance form `A(2,3)` and `B(-4,1)`.

`:. PA=PB`

`=>PA^2=PB^2`

`=>(0-2)^2+(y-3)^2=(0+4)^2+(y-1)^2`

`=>y^2-6y+9+4=y^2-2y+1+16`

`=>-6y+2y=1+16-9-4`

`=>-4y=4`

`=>y=-1`

Thus, the required point is `P(0,-1).`

---

---