The line segment joining the points $A(2, 1)$ and $B(5, -8)$ 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 is divided into three equal parts. Since point $P$ is nearer to $A$, it divides the line segment $AB$ in the ratio $1:2$. This means the distance from $A$ to $P$ is one-third of the total length of $AB$, and the distance from $P$ to $B$ is two-thirds of the total length.

Step 2: Apply Section Formula to find coordinates of P

We use the section formula to find the coordinates of point $P$. Given $A(x_1, y_1) = (2, 1)$ and $B(x_2, y_2) = (5, -8)$, and the ratio $m:n = 1:2$. Substitute these values into the formula to find the $x$ and $y$ 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 location of point $P$ on the coordinate plane.

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

The problem states that point $P(3, -2)$ lies on the line $2x - y + k = 0$. To find the value of $k$, we substitute the $x$ and $y$ coordinates of $P$ into the equation of the line.

Step 5: Solve for k

Simplify the equation by performing the multiplication and addition. Then, isolate $k$ to find its value. This gives us the final answer for the unknown constant $k$.

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]
• Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by the point (-1, 6).
• 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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