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

Hint:

The direction ratios of the two lines are \((p, 1, r)\) and \((p^{\prime}, 1, r^{\prime})\). Their dot product must be zero, which gives \(pp^{\prime} + rr^{\prime} = -1\).

Answer 1

The Correct Option is D

Step 1: Convert the lines to symmetric form:
From $x = py + q$ we get $\dfrac{x - q}{p} = y$. From $z = ry + s$ we get $\dfrac{z - s}{r} = y$.
So the first line is $\dfrac{x - q}{p} = \dfrac{y}{1} = \dfrac{z - s}{r}$, with direction ratios $(p, 1, r)$.
The second line is $\dfrac{x - q^{\prime}}{p^{\prime}} = \dfrac{y}{1} = \dfrac{z - s^{\prime}}{r^{\prime}}$, with direction ratios $(p^{\prime}, 1, r^{\prime})$.

Step 2: Test the claim in (A):
For perpendicular lines, $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$.
Here that reads $pp^{\prime} + 1 + rr^{\prime} = 0$, so $pp^{\prime} + rr^{\prime} = -1$.
The given value is $+1$, so the Assertion is false. For example, $p = 1$, $r = 0$, $p^{\prime} = 1$, $r^{\prime} = 0$ gives $pp^{\prime} + rr^{\prime} = 1$, but the dot product $1 + 1 + 0 = 2$ is not zero.

Step 3: Test the claim in (R):
The angle between two lines is the angle between their direction vectors. If it is a right angle, then $\cos\theta = 0$.
That means $\vec{b}_1 \cdot \vec{b}_2 = |\vec{b}_1||\vec{b}_2|\cos\theta = 0$. So the Reason is true.

Step 4: Match with the options:
A false Assertion with a true Reason is option (D).

Final Answer:
The correct choice is option (D). \[ \boxed{\text{(D)}} \]