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A line is drawn from the point P(1,1,1)...

A line is drawn from the point P(1,1,1)and perpendicular to a line with direction ratios, (1,1,1) to intersect the plane `x+2y+3z=4` at Q. The locus of point Q is

A

`x/1=(y-5)/-2=(z-+2)/1`

B

`x/-2=(y-5)/1=(z+2)/1`

C

`x=y=z`

D

none of these

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we need to find the locus of the point Q, which is the intersection of a line drawn from point P(1, 1, 1) perpendicular to a line with direction ratios (1, 1, 1) and intersects the plane given by the equation \(x + 2y + 3z = 4\). ### Step 1: Define the Line from Point P The line drawn from point P(1, 1, 1) in the direction of the direction ratios (1, 1, 1) can be represented parametrically as: \[ (x, y, z) = (1, 1, 1) + t(1, 1, 1) \] where \(t\) is a parameter. Thus, we have: \[ x = 1 + t, \quad y = 1 + t, \quad z = 1 + t \] ### Step 2: Find the Intersection with the Plane We substitute the parametric equations of the line into the equation of the plane \(x + 2y + 3z = 4\): \[ (1 + t) + 2(1 + t) + 3(1 + t) = 4 \] This simplifies to: \[ 1 + t + 2 + 2t + 3 + 3t = 4 \] \[ 6 + 6t = 4 \] \[ 6t = 4 - 6 \] \[ 6t = -2 \quad \Rightarrow \quad t = -\frac{1}{3} \] ### Step 3: Find the Coordinates of Point Q Now substituting \(t = -\frac{1}{3}\) back into the parametric equations to find the coordinates of point Q: \[ x = 1 - \frac{1}{3} = \frac{2}{3}, \quad y = 1 - \frac{1}{3} = \frac{2}{3}, \quad z = 1 - \frac{1}{3} = \frac{2}{3} \] Thus, the coordinates of point Q are: \[ Q\left(\frac{2}{3}, \frac{2}{3}, \frac{2}{3}\right) \] ### Step 4: Find the Locus of Point Q Since the line PQ is always perpendicular to the line with direction ratios (1, 1, 1), we can express the coordinates of point Q in terms of a variable \(k\): Let: \[ Q(x, y, z) = \left(k, k, k\right) \] Substituting into the plane equation: \[ k + 2k + 3k = 4 \] \[ 6k = 4 \quad \Rightarrow \quad k = \frac{2}{3} \] Thus, the locus of point Q can be expressed as: \[ x = y = z = \frac{2}{3} \] This indicates that point Q lies on the line where \(x = y = z\). ### Conclusion The locus of point Q is given by the equation: \[ x = y = z \]
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