coordinates of a point P lying on the line x-2/4 = y-1/3 = z+1/2 such that the line joining the point P and A(1,3,5) is parallel to the plane 4x-2y+z=0
Answers
To find the coordinates of a point P lying on the line such that the line joining the point P and A(1,3,5) is parallel to the plane 4x-2y+z=0, we can use the following steps:
First, we need to find the direction ratios of the line. We can use the point-slope form of a line, where the direction ratios are the coefficients of x, y, and z. From the given equation we can see that the direction ratios of the line are [4, 3, -2].
Next, we need to find the direction vector of the line joining the point P and A(1,3,5). The direction vector is the vector pointing from A to P. So we can subtract the coordinates of A from the coordinates of P to find the direction vector.
We know that the line joining the point P and A(1,3,5) is parallel to the plane 4x-2y+z=0. So the direction vector of the line should be parallel to the normal vector of the plane. The normal vector of the plane is [4, -2, 1].
We know that the direction vector is parallel to the normal vector, so we can set the direction vector equal to the normal vector and solve for the coordinates of P.
we know that the direction vector is equal to the normal vector, so we can set the direction vector [4, 3, -2] equal to the normal vector [4, -2, 1]
Now we know that the direction vector = normal vector,
so we can use the point A(1,3,5) and the direction vector to find the point P.
We can represent the point P as P(x,y,z)
x = 1 + 4t
y = 3 + 3t
z = 5 - 2t
where t is any real number, we can choose any value for t.
So the point P can be any point that lies on the line x = 1 + 4t, y = 3 + 3t, z = 5 - 2t
So, the point P can be any point on the line x-2/4 = y-1/3 = z+1/2 such that the line joining the point P and A(1,3,5) is parallel to the plane 4x-2y+z=0
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