The line segment joining the points A(3, 2) and B(6, -7) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line $2x - y + k = 0$ find the value of k.

Step 1: Understand Trisection and Section Formula

When a line segment is trisected, it means it is divided into three equal parts. Since P is nearer to A, point P divides the line segment AB in the ratio 1:2. This means the distance AP is one-third of AB, and PB is two-thirds of AB.

Step 2: Apply Section Formula to find coordinates of P

We use the section formula to find the coordinates of point P. Here, $(x_1, y_1) = (3, 2)$, $(x_2, y_2) = (6, -7)$, and the ratio $m:n = 1:2$. Substitute these values into the formula to find the coordinates of P.

Step 3: Calculate coordinates of P

Perform the arithmetic operations to simplify the expressions for the x and y coordinates of P. This gives us the exact coordinates of point P.

Step 4: Substitute P's coordinates into the line equation

The problem states that point P lies on the line $2x - y + k = 0$. This means that the coordinates of P must satisfy the equation of the line. Substitute the x and y coordinates of P into the line equation.

Step 5: Solve for k

Simplify the equation by performing the multiplication and addition. Then, isolate $k$ to find its value. This will give us the final answer.

More from this chapter

• Q7. If A(a, 0), B(1, 1) and C(0, b) form a triangle, right angled at B when joined, then establish a relation between a and b. [2 Marks]
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• Find a point on the y-axis that is at the same distance from the points (-5, 2) and (9, -2).
• Find the coordinates of the point which divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1 internally.
• Find the relation between x and y if the point (x, y) is equidistant from (7, 1) and (3, 5).
• 1. Find the distance between the following pairs of points: (i) (2, 3), (4, 1) (ii) (-5, 7), (-1, 3) (iii) (a, b), (-a, -b)
• Find the coordinates of the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4).
• Find the ratio in which the line segment joining A(1, -5) and B(-4, 5) is divided by the x-axis, and find the coordinates of the point of division.

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