Geometry

Question : How do you find the equation of the circle passing through three points (0,0), (2,1),(0,3) ? Answer : Step 1 : Let equation of the circle is \[x^2+y^2+2gx+2fy+c=0\]…

Question : Does the equation 3x^2+3y^2+5x-6y+1=0 represent a circle ? If yes finds it's radius and center. Answer : Given equation \[3x^2+3y^2+5x-6y+1=0\] Step 1 : Convert this equation into the…

Question : Find the equation of the plane through (1 2 3) parallel to the plane 4x+5y-3z=7 Solution : Let equation of the plane is ax+by+cz+d=0                                                                             (1) Since this plane…

Question: Find the equation of the plane containing two lines (x-1)/2=(y-2)/3=(z-3)/4 and (x-2)/3=(y-3)/4=(z-4)/5 . Solution : First we shall find direction vector of the lines \[\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\] and \[\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\] are \[\vec{d_1}=2\hat{i}+3\hat{j}+4\hat{k},\…

how do you prove that two lines in 3D are coplanar? Apply that to Prove the following question Question : Prove that the lines (x-1)/2=(y-2)/3=(z-3)/4 and (x-2)/3=(y-3)/4=(z-4)/5 are coplanar. Solution:…

Question : Find the direction cosines of a line which makes equal angles with the coordinate axes. Solution : If a line makes angles α , β , γ with…

Question : If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its direction cosines. Solution: If a line makes angles α , β…