Let P(x,3) divide the line segment joining the points A(−4,2) and B(3,6) in the ratio k:1

∴ Coordinates of P is
(‌

m1x2+m2x1

m1+m2

,‌

m1y2+m2y1

m1+m2

) =(‌

3k−4

k+1

,‌

6k+2

k+1

)
But coordinate of P is (x,3)
⇒‌‌‌

6k+2

k+1

‌=3
6k+2‌=3k+3
3k‌=1⇒k=‌

1

3

∴ The required ratio is ‌

1

3

:1 i.e., 1:3 (internally)
(i) Now x=‌

3k−4

k+1

Putting k=‌

1

3

, we get
x‌=‌

3× 1 3 −4

1 3 +1

‌=‌

1−4

1+3 3

=‌

−3

4∕3

=‌

−9

4

(ii) ∴ Coordinate of P is (‌

−9

4

,3)
‌ Length of AP ‌‌=√(x2−x1)2+(y2−y1)2
‌=√(−‌

9

4

+4)2+(3−2)2.
‌=√(‌

−9+16

4

)2+(1)2
‌=√‌

49

16

+1=√‌

49+16

16

‌=√‌

65

16

=‌

√65

4