Question

In what ratio is the line segment joining the points A(- 6, 15) and B(3, 5) is divided by the y-axis internally ?
1. -3 : 1
2. 2 : 1
3. 1 : -3
4. 1 : 2

Answer 1

Correct Answer - Option 2 : 2 : 1

Concept:

 Let A (x1, y1) and B (x2, y2) be the two given points and the point P (x, y) divide the line joining the points A and B in the ratio m : n, then

The point of internal division is given as:

 \(\left( {x,\;y} \right) = \left( {\frac{{m{x_2} + n{x_1}}}{{m + n}},\frac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\)

Calculation:

Let the y-axis divides the line joining the points  A(- 6, 15) and B(3, 5) in the ratio m : 1.

Let C be the point of intersection.

As we know that, the point  internal division is given by: 

\(\left( {x,\;y} \right) = \left( {\frac{{m{x_2} + n{x_1}}}{{m + n}},\frac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\)

\(⇒ C = \left( {\frac{{3m - 6}}{{m + 1}},\frac{{5m + 15}}{{m + 1}}} \right)\)

C is the point of division i.e C lies on the y-axis and the equation of the y-axis is x = 0.

So, the point C will satisfy the equation x = 0

⇒ 3m - 6 = 0

⇒ m = 2

So, the required ratio is = 2 : 1

The point of external division is given as:

 \(\left( {x,y} \right) = \left( {\frac{{m{x_2} - n{x_1}}}{{m - n}},\frac{{m{y_2} - n{y_1}}}{{m - n}}} \right)\)

Note: If P is the mid-point of line segment AB, then \(P\left( {x,\;y} \right) = \left( {\frac{{{x_1} + {x_2}}}{2},\frac{{{y_1} + {y_2}}}{2}} \right)\)