121147 If two distinct point \(Q, R\), lie on the line of intersection of the planes \(-x+2 y-z=0\) and \(3 x\) \(-5 y+2 z=0\) and \(P Q=P R=\sqrt{18}\) where the point \(P\) is \((1,-2,3)\), then the area of the triangle \(\mathrm{PQR}\) is equal to

121148 The distance of the point \(P(4,6-2)\) from th line passing the point \((-3,2,3)\) and parallel to line with direction ratios \(3,3,-1\) is equal to:

121149 Consider the lines \(L_1\) and \(L_2\) given by
\(L_1: \frac{x-1}{2}=\frac{y-3}{1}=\frac{z-2}{2}\)
\(L_2: \frac{x-2}{2}=\frac{y-2}{2}=\frac{z-3}{3}\)
A line \(\mathrm{L}_3\) having direction ratios \(1,-1,-2\), intersects \(L_1\) and \(L_2\) at the point \(P\) and \(Q\) respectively. The length of the line segment \(P Q\) is

121150 If a plane passes through the point \((-1, k, 0),(2\), \(k,-1),(1,1,2)\) and is parallel to the line
\(\frac{x-1}{1}=\frac{2 y+1}{2}=\frac{z+1}{-1}\), then the value of \(\frac{\mathbf{k}^2+\mathbf{1}}{(\mathbf{k}-\mathbf{1})(\mathrm{k}-\mathbf{2})}\) is

121147 If two distinct point \(Q, R\), lie on the line of intersection of the planes \(-x+2 y-z=0\) and \(3 x\) \(-5 y+2 z=0\) and \(P Q=P R=\sqrt{18}\) where the point \(P\) is \((1,-2,3)\), then the area of the triangle \(\mathrm{PQR}\) is equal to

121148 The distance of the point \(P(4,6-2)\) from th line passing the point \((-3,2,3)\) and parallel to line with direction ratios \(3,3,-1\) is equal to:

121149 Consider the lines \(L_1\) and \(L_2\) given by
\(L_1: \frac{x-1}{2}=\frac{y-3}{1}=\frac{z-2}{2}\)
\(L_2: \frac{x-2}{2}=\frac{y-2}{2}=\frac{z-3}{3}\)
A line \(\mathrm{L}_3\) having direction ratios \(1,-1,-2\), intersects \(L_1\) and \(L_2\) at the point \(P\) and \(Q\) respectively. The length of the line segment \(P Q\) is

121150 If a plane passes through the point \((-1, k, 0),(2\), \(k,-1),(1,1,2)\) and is parallel to the line
\(\frac{x-1}{1}=\frac{2 y+1}{2}=\frac{z+1}{-1}\), then the value of \(\frac{\mathbf{k}^2+\mathbf{1}}{(\mathbf{k}-\mathbf{1})(\mathrm{k}-\mathbf{2})}\) is

121147 If two distinct point \(Q, R\), lie on the line of intersection of the planes \(-x+2 y-z=0\) and \(3 x\) \(-5 y+2 z=0\) and \(P Q=P R=\sqrt{18}\) where the point \(P\) is \((1,-2,3)\), then the area of the triangle \(\mathrm{PQR}\) is equal to

121148 The distance of the point \(P(4,6-2)\) from th line passing the point \((-3,2,3)\) and parallel to line with direction ratios \(3,3,-1\) is equal to:

121149 Consider the lines \(L_1\) and \(L_2\) given by
\(L_1: \frac{x-1}{2}=\frac{y-3}{1}=\frac{z-2}{2}\)
\(L_2: \frac{x-2}{2}=\frac{y-2}{2}=\frac{z-3}{3}\)
A line \(\mathrm{L}_3\) having direction ratios \(1,-1,-2\), intersects \(L_1\) and \(L_2\) at the point \(P\) and \(Q\) respectively. The length of the line segment \(P Q\) is

121150 If a plane passes through the point \((-1, k, 0),(2\), \(k,-1),(1,1,2)\) and is parallel to the line
\(\frac{x-1}{1}=\frac{2 y+1}{2}=\frac{z+1}{-1}\), then the value of \(\frac{\mathbf{k}^2+\mathbf{1}}{(\mathbf{k}-\mathbf{1})(\mathrm{k}-\mathbf{2})}\) is

121147 If two distinct point \(Q, R\), lie on the line of intersection of the planes \(-x+2 y-z=0\) and \(3 x\) \(-5 y+2 z=0\) and \(P Q=P R=\sqrt{18}\) where the point \(P\) is \((1,-2,3)\), then the area of the triangle \(\mathrm{PQR}\) is equal to

121148 The distance of the point \(P(4,6-2)\) from th line passing the point \((-3,2,3)\) and parallel to line with direction ratios \(3,3,-1\) is equal to:

121149 Consider the lines \(L_1\) and \(L_2\) given by
\(L_1: \frac{x-1}{2}=\frac{y-3}{1}=\frac{z-2}{2}\)
\(L_2: \frac{x-2}{2}=\frac{y-2}{2}=\frac{z-3}{3}\)
A line \(\mathrm{L}_3\) having direction ratios \(1,-1,-2\), intersects \(L_1\) and \(L_2\) at the point \(P\) and \(Q\) respectively. The length of the line segment \(P Q\) is

121150 If a plane passes through the point \((-1, k, 0),(2\), \(k,-1),(1,1,2)\) and is parallel to the line
\(\frac{x-1}{1}=\frac{2 y+1}{2}=\frac{z+1}{-1}\), then the value of \(\frac{\mathbf{k}^2+\mathbf{1}}{(\mathbf{k}-\mathbf{1})(\mathrm{k}-\mathbf{2})}\) is