121375 Let \(P\) be the point of intersection of the line \(\frac{x+3}{3}=\frac{y+2}{1}=\frac{1-z}{2}\) and the plane \(x+y+z=2\). If the distance of the point \(P\) from the plane \(3 x\) \(-4 y+12 z=32\) is \(q\), then \(q\) and \(2 q\) are the roots of the equation

121378 Let the line \(\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}\) intersect the lines \(\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}\) and \(\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}\) at the points \(A\) and \(B\) respectively. Then the distance of the mid-point of the line segment \(A B\) from the plane \(2 x-2 y+z=14\) is

121380 A plane \(\mathrm{E}\) is perpendicular to the two planes \(2 \mathrm{x}\) \(-2 y+z=0\) and \(x-y+2 z=4\), and passes through the point \(P(1,-1,1)\). If the distance of the plane \(E\) from the point \(Q(a, a, 2)\) is \(3 \sqrt{2}\), then \((\mathrm{PQ})^2\) is equal to

121387 The length of perpendicular from point \(\hat{i}+2 \hat{j}+3 \hat{k}\) to the line \(\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}\) is

121375 Let \(P\) be the point of intersection of the line \(\frac{x+3}{3}=\frac{y+2}{1}=\frac{1-z}{2}\) and the plane \(x+y+z=2\). If the distance of the point \(P\) from the plane \(3 x\) \(-4 y+12 z=32\) is \(q\), then \(q\) and \(2 q\) are the roots of the equation

121378 Let the line \(\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}\) intersect the lines \(\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}\) and \(\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}\) at the points \(A\) and \(B\) respectively. Then the distance of the mid-point of the line segment \(A B\) from the plane \(2 x-2 y+z=14\) is

121380 A plane \(\mathrm{E}\) is perpendicular to the two planes \(2 \mathrm{x}\) \(-2 y+z=0\) and \(x-y+2 z=4\), and passes through the point \(P(1,-1,1)\). If the distance of the plane \(E\) from the point \(Q(a, a, 2)\) is \(3 \sqrt{2}\), then \((\mathrm{PQ})^2\) is equal to

121387 The length of perpendicular from point \(\hat{i}+2 \hat{j}+3 \hat{k}\) to the line \(\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}\) is

121375 Let \(P\) be the point of intersection of the line \(\frac{x+3}{3}=\frac{y+2}{1}=\frac{1-z}{2}\) and the plane \(x+y+z=2\). If the distance of the point \(P\) from the plane \(3 x\) \(-4 y+12 z=32\) is \(q\), then \(q\) and \(2 q\) are the roots of the equation

121378 Let the line \(\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}\) intersect the lines \(\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}\) and \(\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}\) at the points \(A\) and \(B\) respectively. Then the distance of the mid-point of the line segment \(A B\) from the plane \(2 x-2 y+z=14\) is

121380 A plane \(\mathrm{E}\) is perpendicular to the two planes \(2 \mathrm{x}\) \(-2 y+z=0\) and \(x-y+2 z=4\), and passes through the point \(P(1,-1,1)\). If the distance of the plane \(E\) from the point \(Q(a, a, 2)\) is \(3 \sqrt{2}\), then \((\mathrm{PQ})^2\) is equal to

121387 The length of perpendicular from point \(\hat{i}+2 \hat{j}+3 \hat{k}\) to the line \(\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}\) is

121375 Let \(P\) be the point of intersection of the line \(\frac{x+3}{3}=\frac{y+2}{1}=\frac{1-z}{2}\) and the plane \(x+y+z=2\). If the distance of the point \(P\) from the plane \(3 x\) \(-4 y+12 z=32\) is \(q\), then \(q\) and \(2 q\) are the roots of the equation

121378 Let the line \(\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}\) intersect the lines \(\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}\) and \(\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}\) at the points \(A\) and \(B\) respectively. Then the distance of the mid-point of the line segment \(A B\) from the plane \(2 x-2 y+z=14\) is

121380 A plane \(\mathrm{E}\) is perpendicular to the two planes \(2 \mathrm{x}\) \(-2 y+z=0\) and \(x-y+2 z=4\), and passes through the point \(P(1,-1,1)\). If the distance of the plane \(E\) from the point \(Q(a, a, 2)\) is \(3 \sqrt{2}\), then \((\mathrm{PQ})^2\) is equal to

121387 The length of perpendicular from point \(\hat{i}+2 \hat{j}+3 \hat{k}\) to the line \(\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}\) is