Q. 185.0( 1 Vote )

# Find the length and the foot of perpendicular from the point to the plane 2x – 2y + 4z + 5 = 0.

Answer :

It is given that,

The point is .

The plane is 2x – 2y + 4z + 5 = 0

We need to find the foot of the perpendicular from the Point P to the equation of the given plane.

Also, we need to find the distance from the point P to the plane.

Let Q be the foot of the perpendicular from the point .

Let the point Q be Q(x_{1}, y_{1}, z_{1}).

We know the direction ratio of any line segment PQ, where P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}), is given by (x_{2} – x_{1}, y_{2} – y_{1}, z_{2} – z_{1}).

So, the direction ratio of PQ is given by

Direction ratio of PQ = (x_{1} – 1, y_{1} – 3/2, z_{1} – 2)

Now, let be the normal to the plane 2x – 2y + 4z + 5 = 0.

is obviously parallel to the since, a normal is an object such as a line or vector that is perpendicular to a given object.

Direction ratio simply states the number of units to move along each axis.

For any plane, ax + by + cz = d

a, b and c are normal vectors to the plane.

And therefore, the direction ratios are (a, b, c).

So, direction ratio of = (2, -2, 4) for plane 2x – 2y + 4z + 5 = 0.

Cartesian equation of line PQ, where P(1, 3/2, 2) and Q(x_{1}, y_{1}, z_{1}) is

Let us find any point on this line.

From

⇒ x_{1} – 1 = 2λ

⇒ x_{1} = 2λ + 1

From

From

⇒ z_{1} – 2 = 4λ

⇒ z_{1} = 4λ + 2

Any point on the line is (2λ + 1, 3/2 – 2λ, 4λ + 2).

This point is Q.

Q(x_{1}, y_{1}, z_{1}) = Q(2λ + 1, 3/2 – 2λ, 4λ + 2) …(i)

And it was assumed to be lying on the given plane. So, substitute x_{1}, y_{1} and z_{1} in the plane equation, we get

2x_{1} – 2y_{1} + 4z_{1} + 5 = 0

Simplifying it to find the value of λ.

⇒ 4λ + 2 – 3 + 4λ + 16λ + 8 + 5 = 0

⇒ 4λ + 4λ + 16λ + 2 – 3 + 8 + 5 = 0

⇒ 24λ + 12 = 0

⇒ 24λ = -12

Since, Q is the foot of the perpendicular from the point P.

The, substitute the value of λ in equation (i),

Also, we need to find .

Where,

P = (1, 3/2, 2)

Q = (0, 5/2, 0)

is found as,

**Thus, foot of the perpendicular from the given point to the plane is (0, 5/2,0) and the distance is √6 units.**

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