Question

Assertion (A) : Lines given by \(x = py + q\), \(z = ry + s\) and \(x = p^{\prime}y + q^{\prime}\), \(z = r^{\prime}y + s^{\prime}\) are perpendicular to each other when \(pp^{\prime} + rr^{\prime} = 1\).

Reason (R) : Two lines \(\vec{r} = \vec{a}_1 + \lambda\vec{b}_1\) and \(\vec{r} = \vec{a}_2 + \mu\vec{b}_2\) are perpendicular to each other if \(\vec{b}_1 \cdot \vec{b}_2 = 0\).

• A. Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true and Reason (R) is false.
• D. Assertion (A) is false and Reason (R) is true.

Hint:

The direction ratios of the two lines in (A) are (p, 1, r) and (p-dash, 1, r-dash). Perpendicular lines need their dot product to be 0.

Answer 1

The Correct Option is D

Step 1: Plan.
We test the Assertion with a small numerical example. Then we judge the Reason from the dot product rule.

Step 2: Write the lines in vector form.
Put $y = t$. The first line becomes $(x, y, z) = (q, 0, s) + t\,(p, 1, r)$.
So the direction vector is $\vec{b}_1 = p\hat{i} + \hat{j} + r\hat{k}$.
Similarly the second direction vector is $\vec{b}_2 = p^{\prime}\hat{i} + \hat{j} + r^{\prime}\hat{k}$.

Step 3: Compute the dot product.
\[ \vec{b}_1 \cdot \vec{b}_2 = pp^{\prime} + 1 + rr^{\prime} \]
Perpendicular lines need this to be $0$. So we need $pp^{\prime} + rr^{\prime} = -1$.

Step 4: Test the Assertion with numbers.
Take $p = p^{\prime} = 1$ and $r = r^{\prime} = 0$. Then $pp^{\prime} + rr^{\prime} = 1$, as in the Assertion.
But $\vec{b}_1 \cdot \vec{b}_2 = 1 + 1 + 0 = 2 \neq 0$. The lines are not perpendicular.
So the Assertion is false.

Step 5: Judge the Reason.
The Reason says that $\vec{b}_1 \cdot \vec{b}_2 = 0$ gives perpendicular lines. This is the standard rule, so it is true.

Final Answer:
Assertion false, Reason true: option (D). \[ \boxed{\text{(D)}} \]