Question:
Given two points \(Q(3,4)\) and \(R(1,2)\). What is the point \(P(x,y)\) on the line \(2x - y - 1 = 0\) for which \(PQ + PR = QR\) holds?
Given two points \(Q(3,4)\) and \(R(1,2)\). What is the point \(P(x,y)\) on the line \(2x - y - 1 = 0\) for which \(PQ + PR = QR\) holds?
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When a point P satisfies \(PQ + PR = QR\), it lies on segment QR; use parametric representation and line equation to find coordinates.
Updated On: Jul 18, 2026
- \((-3,-7)\)
- \((-2,-5)\)
- \((2,3)\)
- \((4,7)\)
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The Correct Option is C
Solution and Explanation
Step 1: Recognize the condition.
Point \(P\) must lie on line \(2x - y - 1 = 0\) and satisfy \(PQ + PR = QR\), which is the triangle equality implying \(P\) lies on segment \(QR\).
Step 2: Parametric form of QR.
\[ x = 1 + t(3-1) = 1 + 2t, \quad y = 2 + t(4-2) = 2 + 2t, \quad 0 \le t \le 1 \]
Step 3: Apply line condition.
\[ 2x - y - 1 = 0 \implies 2(1+2t) - (2+2t) - 1 = 0 \]
Step 4: Solve for t.
\[ 2 + 4t - 2 - 2t -1 = 0 \implies 2t - 1 = 0 \implies t = \frac{1}{2} \]
Step 5: Find coordinates of P.
\[ x = 1 + 2 \cdot \frac{1}{2} = 2, \quad y = 2 + 2 \cdot \frac{1}{2} = 3 \]
Step 6: Final conclusion.
Hence, the point is \[ \boxed{(2,3)} \]
Point \(P\) must lie on line \(2x - y - 1 = 0\) and satisfy \(PQ + PR = QR\), which is the triangle equality implying \(P\) lies on segment \(QR\).
Step 2: Parametric form of QR.
\[ x = 1 + t(3-1) = 1 + 2t, \quad y = 2 + t(4-2) = 2 + 2t, \quad 0 \le t \le 1 \]
Step 3: Apply line condition.
\[ 2x - y - 1 = 0 \implies 2(1+2t) - (2+2t) - 1 = 0 \]
Step 4: Solve for t.
\[ 2 + 4t - 2 - 2t -1 = 0 \implies 2t - 1 = 0 \implies t = \frac{1}{2} \]
Step 5: Find coordinates of P.
\[ x = 1 + 2 \cdot \frac{1}{2} = 2, \quad y = 2 + 2 \cdot \frac{1}{2} = 3 \]
Step 6: Final conclusion.
Hence, the point is \[ \boxed{(2,3)} \]
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