The image of point \(P(x, y, z)\) with respect to the line:
\[
\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3},
\]
is \(P'(1, 0, 7)\). Find the coordinates of point \(P\).
Show Hint
Solution and Explanation
Step 1: Parametric equation of the line. The given line can be expressed in parametric form as: \[ x = t, \quad y = 1 + 2t, \quad z = 2 + 3t, \] where \(t\) is a parameter.
Step 2: Midpoint of \(P\) and \(P'\). The image \(P'(1, 0, 7)\) of \(P(x, y, z)\) with respect to the given line implies that the midpoint \(M\) of \(P\) and \(P'\) lies on the line.
The coordinates of the midpoint \(M\) are: \[ M = \left(\frac{x + 1}{2}, \frac{y + 0}{2}, \frac{z + 7}{2}\right). \]
Step 3: Condition for \(M\) lying on the line. Since \(M\) lies on the line, its coordinates must satisfy the parametric equations of the line. Thus: \[ \frac{x + 1}{2} = t, \quad \frac{y}{2} = 1 + 2t, \quad \frac{z + 7}{2} = 2 + 3t. \]
Step 4: Solve for \(t\). From the first equation: \[ t = \frac{x + 1}{2}. \] Substitute \(t\) into the second equation: \[ \frac{y}{2} = 1 + 2\left(\frac{x + 1}{2}\right). \]
Simplify: \[ \frac{y}{2} = 1 + x + 1 \implies \frac{y}{2} = x + 2 \implies y = 2x + 4. \tag{1} \]
Substitute \(t\) into the third equation: \[ \frac{z + 7}{2} = 2 + 3\left(\frac{x + 1}{2}\right). \]
Simplify: \[ \frac{z + 7}{2} = 2 + \frac{3x + 3}{2} \implies z + 7 = 4 + 3x + 3 \implies z = 3x. \tag{2} \]
Step 5: Use the coordinates of \(P'\) to find \(P\). From the midpoint condition: \[ \frac{x + 1}{2} = t, \quad \frac{y + 0}{2} = 1 + 2t, \quad \frac{z + 7}{2} = 2 + 3t. \]
Substitute \(t = \frac{x + 1}{2}\): \[ \frac{y + 0}{2} = 1 + 2\left(\frac{x + 1}{2}\right). \]
Simplify: \[ \frac{y}{2} = 1 + x + 1 \implies y = 2x + 4. \tag{3} \] Similarly: \[ \frac{z + 7}{2} = 2 + 3\left(\frac{x + 1}{2}\right). \] Simplify: \[ \frac{z + 7}{2} = 2 + \frac{3x + 3}{2} \implies z = 3x. \tag{4} \]
Step 6: Solve for \(P\). We now solve for the coordinates of \(P(x, y, z)\).
Recall the midpoint conditions and verify: From the midpoint: \[ M = \left(\frac{x + 1}{2}, \frac{y}{2}, \frac{z + 7}{2}\right), \] and since \(M\) lies on the line, its coordinates must satisfy: \[ x = t, \quad y = 1 + 2t, \quad z = 2 + 3t. \]
Using the given image \(P'(1, 0, 7)\) and midpoint relationships: \[ \frac{x + 1}{2} = t, \quad \frac{y}{2} = 1 + 2t, \quad \frac{z + 7}{2} = 2 + 3t. \]
Substitute \(t = \frac{x + 1}{2}\) into the second and third equations: 1.
From the second equation: \[ \frac{y}{2} = 1 + 2\left(\frac{x + 1}{2}\right). \] Simplify: \[ \frac{y}{2} = 1 + x + 1 \implies y = 2x + 4. \] 2. From the third equation: \[ \frac{z + 7}{2} = 2 + 3\left(\frac{x + 1}{2}\right). \]
Simplify: \[ \frac{z + 7}{2} = 2 + \frac{3x + 3}{2} \implies z + 7 = 4 + 3x + 3 \implies z = 3x. \] Thus, the coordinates of \(P(x, y, z)\) are determined as: \[ P(x, y, z) = (0, 4, 0). \]
Final Answer: The coordinates of \(P\) are: \[ P(x, y, z) = (0, 4, 0). \]
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 Three Dimensional Geometry Questions
- The distance of point P(a, b, c) from the y-axis is:
- The number of corner points of the feasible region determined by constraints \(x \geq 0\), \(y \geq 0\), \(x + y \geq 4\) is:
- Assertion (A): For the matrix \[ A = \begin{bmatrix} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{bmatrix}, \quad \text{where } \theta \in [0, 2\pi], \] \(|A| \in [2, 4]\).
Reason (R): \(\cos \theta \in [-1, 1], \ \forall \ \theta \in [0, 2\pi].\) Assertion (A): A line in space cannot be drawn perpendicular to \( x \), \( y \), and \( z \) axes simultaneously.
Reason (R): For any line making angles \( \alpha, \beta, \gamma \) with the positive directions of \( x \), \( y \), and \( z \) axes respectively, \[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1. \]- The distance of the point \( P(a, b, c) \) from the y-axis 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