If P and Q are two points on the curve y=2^{x | vedclass

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(A) Let point $P(p, 2^{p+2})$ and $Q(q, 2^{q+2})$.Given $OP = p\hat{i} + 2^{p+2}\hat{j}$ and $OQ = q\hat{i} + 2^{q+2}\hat{j}$.According to the questio

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$10$

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(A) Let point $P(p, 2^{p+2})$ and $Q(q, 2^{q+2})$.Given $OP = p\hat{i} + 2^{p+2}\hat{j}$ and $OQ = q\hat{i} + 2^{q+2}\hat{j}$.According to the question,$OP \cdot \hat{i} = -1$,which implies $p = -1$.Thus,$OP = -\hat{i} + 2^{-1+2}\hat{j} = -\hat{i} + 2\hat{j}$.Also,$OQ \cdot \hat{i} = 2$,which implies $q = 2$.Thus,$OQ = 2\hat{i} + 2^{2+2}\hat{j} = 2\hat{i} + 16\hat{j}$.Now,calculate $OQ - 4OP = (2\hat{i} + 16\hat{j}) - 4(-\hat{i} + 2\hat{j}) = (2+4)\hat{i} + (16-8)\hat{j} = 6\hat{i} + 8\hat{j}$.The magnitude is $|OQ - 4OP| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$.

If $P$ and $Q$ are two points on the curve $y=2^{x+2}$ such that $OP \cdot \hat{i}=-1$ and $OQ \cdot \hat{i}=2$,then the magnitude of $(OQ-4OP)$ is

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