In what ratio is the join 4 3 and 2 6 divided by the x-axis also find the coordinates of the point of intersection?

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

• Point of internal division is given as: \(\left( {x,\;y} \right) = \left( {\frac{{m{x_2} + n{x_1}}}{{m + n}},\frac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\)
• 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 \(P\left( {x,\;y} \right) = \left( {\frac{{{x_1} + {x_2}}}{2},\frac{{{y_1} + {y_2}}}{2}} \right)\)

CALCULATION:

Given: X-axis cuts the line joining the points A(4, 5) and B(-10, - 2) internally

Here, we have to find the point of intersection i.e the point of internal division.

Let C be the point of internal division such that the ratio in which x-axis cuts the line joining the points A(4, 5) and B(-10, - 2) in the ratio m : 1

As we know that, the point  internal division is given by: \(\left( {x,\;y} \right) = \left( {\frac{{m{x_2} + n{x_1}}}{{m + n}},\frac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\)

\(⇒ C = \left( {\frac{{- 10m + 4}}{{m + 1}},\frac{{- 2m + 5}}{{m + 1}}} \right)\)-------------(1)

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

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

⇒ - 2m + 5 = 0 ⋅ (m + 1) = 0

⇒ m = 5/2

By substituting m = 5/2 in the equation (1) we get

⇒ C = (- 6, 0)

Hence, option D is the correct answer.

Find the coordinates of the points which divide the line segment joining A(– 2, 2) and B(2, 8) into four equal parts.

Let P, Q and R be the three points which divide the line-segment joining the points A(-2, 2) and B(2, 8) in four equal parts.

Case I. For point P, we have

Hence, m1 = 1, m2 = 3
x1 = -2, y2 = 2
x2 = 2, y2 = 8Then, coordinates of P are given by

Case II. For point Q, we have

m1 = 2, m2 = 2
x1 = -2, y1 = 2
and    x2 = 2, y2 = 8Then, coordinates of Q are given by

Case III. For point R, we have