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Let P and Q are two points in the xy pla...

Let P and Q are two points in the xy plane on the curve `y = x^(11) - 2x^(7) + 7x^(3) + 11x + 6` such that `vec(OP) cdot hati = 5, vec(OQ) cdot hati = -5`, then the magnitude of `vec(OP) + vec(OQ)` is

A

10

B

12

C

14

D

8

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find the magnitude of the vector sum of points P and Q on the given curve. Let's break down the steps: ### Step 1: Define the Points Let the coordinates of point P be \( P(x_1, y_1) \) and the coordinates of point Q be \( Q(x_2, y_2) \). The vectors from the origin O to these points can be expressed as: \[ \vec{OP} = x_1 \hat{i} + y_1 \hat{j} \] \[ \vec{OQ} = x_2 \hat{i} + y_2 \hat{j} \] ### Step 2: Use Given Conditions According to the problem: \[ \vec{OP} \cdot \hat{i} = x_1 = 5 \] \[ \vec{OQ} \cdot \hat{i} = x_2 = -5 \] ### Step 3: Find y-coordinates Since both points lie on the curve defined by: \[ y = x^{11} - 2x^{7} + 7x^{3} + 11x + 6 \] We can find \( y_1 \) and \( y_2 \) by substituting \( x_1 \) and \( x_2 \) into the curve equation. For point P: \[ y_1 = (5)^{11} - 2(5)^{7} + 7(5)^{3} + 11(5) + 6 \] For point Q: \[ y_2 = (-5)^{11} - 2(-5)^{7} + 7(-5)^{3} + 11(-5) + 6 \] ### Step 4: Calculate \( y_1 \) and \( y_2 \) Calculating \( y_1 \): \[ y_1 = 5^{11} - 2 \cdot 5^{7} + 7 \cdot 5^{3} + 11 \cdot 5 + 6 \] Calculating \( y_2 \): \[ y_2 = -5^{11} + 2 \cdot 5^{7} - 7 \cdot 5^{3} - 11 \cdot 5 + 6 \] ### Step 5: Add the Vectors Now we can add the vectors: \[ \vec{OP} + \vec{OQ} = (x_1 + x_2) \hat{i} + (y_1 + y_2) \hat{j} \] Substituting \( x_1 \) and \( x_2 \): \[ x_1 + x_2 = 5 - 5 = 0 \] Now, we need to find \( y_1 + y_2 \): \[ y_1 + y_2 = \left(5^{11} - 2 \cdot 5^{7} + 7 \cdot 5^{3} + 11 \cdot 5 + 6\right) + \left(-5^{11} + 2 \cdot 5^{7} - 7 \cdot 5^{3} - 11 \cdot 5 + 6\right) \] This simplifies to: \[ y_1 + y_2 = (5^{11} - 5^{11}) + (-2 \cdot 5^{7} + 2 \cdot 5^{7}) + (7 \cdot 5^{3} - 7 \cdot 5^{3}) + (11 \cdot 5 - 11 \cdot 5) + (6 + 6) \] Thus, we have: \[ y_1 + y_2 = 12 \] ### Step 6: Final Vector and Magnitude Now substituting back, we have: \[ \vec{OP} + \vec{OQ} = 0 \hat{i} + 12 \hat{j} \] The magnitude of this vector is: \[ |\vec{OP} + \vec{OQ}| = \sqrt{(0)^2 + (12)^2} = 12 \] ### Conclusion The magnitude of \( \vec{OP} + \vec{OQ} \) is \( 12 \).
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