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 \(\vec{b}_1 \cdot \vec{b}_2 = 0\).
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 \(\vec{b}_1 \cdot \vec{b}_2 = 0\).
Show Hint
- Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
- Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- Assertion (A) is true and Reason (R) is false.
- Assertion (A) is false and Reason (R) is true.
The Correct Option is D
Solution and Explanation
Two lines are perpendicular when the dot product of their direction vectors is zero. We first check the Reason, then use it on the Assertion.
Step 2: Check the Reason (R).
The lines \(\vec{r} = \vec{a}_1 + \lambda\vec{b}_1\) and \(\vec{r} = \vec{a}_2 + \mu\vec{b}_2\) have direction vectors \(\vec{b}_1\) and \(\vec{b}_2\).
The angle between the lines is the angle between \(\vec{b}_1\) and \(\vec{b}_2\). It is \(90^{\circ}\) exactly when \(\vec{b}_1 \cdot \vec{b}_2 = 0\).
So (R) is TRUE.
Step 3: Find the direction ratios of the lines in (A).
Take \(y\) as the free variable. The first line is \(x = py + q\), \(z = ry + s\). In symmetric form: \[ \frac{x - q}{p} = \frac{y}{1} = \frac{z - s}{r} \]
So its direction ratios are \(p, 1, r\).
In the same way, the second line has direction ratios \(p^{\prime}, 1, r^{\prime}\).
Step 4: Apply the condition of perpendicularity to (A).
The lines are perpendicular when \[ p p^{\prime} + (1)(1) + r r^{\prime} = 0 \]
This gives \(pp^{\prime} + rr^{\prime} = -1\), not \(+1\).
So the Assertion, which says \(pp^{\prime} + rr^{\prime} = 1\), is FALSE.
Step 5: Pick the option.
(A) is false and (R) is true. Options (A), (B) and (C) all say that the Assertion is true, so they are wrong. Option (D) is correct.
Final Answer:
Assertion is false and Reason is true. The correct choice is option (D). \[ \boxed{\text{(D)}} \]
Top CBSE CLASS XII Mathematics Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
Top CBSE CLASS XII Equation of a Line in Space Questions
- Find the co-ordinates of the point on line \( x = \frac{y-1}{2} = \frac{z-2}{3} \) whose \( y \) coordinate is 3 times the \( x \) coordinate.
- Find the equation of a line (in vector and cartesian form) that passes through the point of intersection of lines \( \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} \) and \( \frac{x - 4}{5} = \frac{y - 1}{2} = z \) and is parallel to the vector \( 3\hat{i} + 2\hat{j} - 8\hat{k} \).
The length of perpendicular drawn from point \( (2, 5, 7) \) on line \( \dfrac{x}{1} = \dfrac{y}{0} = \dfrac{z}{0} \) is
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 \( \vec{b}_1 \cdot \vec{b}_2 = 0 \).
The length of perpendicular drawn from the point \((1, 2, 3)\) on line \(\dfrac{x}{0} = \dfrac{y}{1} = \dfrac{z}{0}\) is
Top CBSE CLASS XII Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is
- A capillary tube of radius r is dipped inside a large vessel of water. The mass of water raised above water level is M. If the radius of capillary is doubled, the mass of water inside capillary will be
- A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is
- A circular disc is rotating about its own axis at uniform angular velocity \(\omega.\) The disc is subjected to uniform angular retardation by which its angular velocity is decreased to \(\frac {\omega}{2}\) during 120 rotations. The number of rotations further made by it before coming to rest is