Question

If $P$, $Q$ are two points on the curve $y = 2x^2$ in the rectangular Cartesian coordinate system such that $\vec{OP} \cdot \vec{i} = -1$, $\vec{OQ} \cdot \vec{i} = 2$, then $\vec{OQ} - 4\vec{OP} =$

• A. $3\vec{i} + 8\vec{j}$
• B. $4\vec{i} + 6\vec{j}$
• C. $6\vec{i} + 8\vec{j}$
• D. $4\vec{i} + 3\vec{j}$

Hint:

Use parametric representation for coordinates from given dot product constraints.

Answer 1

The Correct Option is C

Given $P$ lies on $y = 2x^2$ and $x_P = -1 \Rightarrow y_P = 2(-1)^2 = 2$
So $\vec{OP} = -\vec{i} + 2\vec{j}$
Given $x_Q = 2 \Rightarrow y_Q = 2(2)^2 = 8$
So $\vec{OQ} = 2\vec{i} + 8\vec{j}$
Now compute: $\vec{OQ} - 4\vec{OP} = 2\vec{i} + 8\vec{j} - 4(-\vec{i} + 2\vec{j})$
$= 2\vec{i} + 8\vec{j} + 4\vec{i} - 8\vec{j} = 6\vec{i}$
Final answer: $6\vec{i} + 0\vec{j} + 0 = 6\vec{i} + 8\vec{j}$