A line segment joining the points P(-5, 11) and Q is divided internally by the point M(2, -3) such that PM : MQ = 7 : 2. The coordinates of Q are :
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Step-by-Step Solution
Step 1: Understand the Section Formula
The section formula is used to find the coordinates of a point that divides a line segment in a given ratio. Here, point M divides the line segment PQ in the ratio m:n=7:2. We are given the coordinates of P and M, and we need to find the coordinates of Q.
Step 2: Identify Given Values
Let the coordinates of P be (x1,y1), the coordinates of M be (x, y), and the coordinates of Q be (x2,y2). The given ratio is m:n=7:2. We need to find (x2,y2).
Step 3: Apply Section Formula for x-coordinate
Substitute the known values into the section formula for the x -coordinate. We have x=2, m=7, n=2, and x1=−5. We will solve this equation for x2.
Step 4: Solve for x2
Simplify the equation and solve for x2. Multiply both sides by 9, then add 10 to both sides, and finally divide by 7 to find the value of x2.
Step 5: Apply Section Formula for y-coordinate
Similarly, substitute the known values into the section formula for the y -coordinate. We have y=−3, m=7, n=2, and y1=11. We will solve this equation for y2.
Step 6: Solve for y2
Simplify the equation and solve for y2. Multiply both sides by 9, then subtract 22 from both sides, and finally divide by 7 to find the value of y2.
Step 7: State the Coordinates of Q
Combining the calculated x2 and y2 values, the coordinates of point Q are (4,−7).