In what ratio is the segment joining the points (2,5) and (−6,−10) divided by the y-axis?
- A3:1
- B1:3
- C2:5
- D5:2
Solution & Step-by-step Explanation
Let the y-axis divide the line segment joining A(2,5) and B(−6,−10) in the ratio k:1 at a point P.
Since the point P lies on the y-axis, its x-coordinate must be 0.
Using the section formula for the x-coordinate:
x=
k+1
k⋅x
2
+1⋅x
1
Here, x
1
=2 and x
2
=−6. Setting x=0:
0=
k+1
k(−6)+1(2)
0=−6k+2
6k=2
k=
6
2
=
3
1
Thus, the ratio k:1 is
3
1
:1, which simplifies to 1:3.
Since the point P lies on the y-axis, its x-coordinate must be 0.
Using the section formula for the x-coordinate:
x=
k+1
k⋅x
2
+1⋅x
1
Here, x
1
=2 and x
2
=−6. Setting x=0:
0=
k+1
k(−6)+1(2)
0=−6k+2
6k=2
k=
6
2
=
3
1
Thus, the ratio k:1 is
3
1
:1, which simplifies to 1:3.