Visual SolutionEasy

Question

The distance of the point

(2, 3, -5)

from the plane

x + 2y - 2z = 9

is:

Solution Path

Point-to-plane distance formula with a=1, b=2, c=-2. Numerator = |2+6+10-9| = 9, denominator = 3. Distance = 3.

01Question Setup

1/4

Find the distance of point

(2, 3, -5)

from the plane

x + 2y - 2z = 9

d = \;?

02Distance Formula

2/4

Use the point-to-plane distance formula:

d = \frac{|ax_1 + by_1 + cz_1 - d|}{\sqrt{a^2+b^2+c^2}}

. Here

a=1, b=2, c=-2, d=9

d = \frac{|\text{numerator}|}{\sqrt{1+4+4}}

03Substitute and ComputeKEY INSIGHT

3/4

Numerator:

|1(2)+2(3)+(-2)(-5)-9| = |2+6+10-9| = 9

. Denominator:

\sqrt{9} = 3

d = \frac{9}{3} = 3

04Final Answer

4/4

Distance

= 3

. Sanity check: point gives

2+6+10=18

, plane constant is

9

, difference

= 9

, divided by

3

gives

3

\boxed{3}

Concepts from this question2 concepts unlocked

★

Equation of a Plane

STANDARD

A plane with normal direction (a, b, c) passing through (x1, y1, z1) has the equation a(x-x1) + b(y-y1) + c(z-z1) = 0, which simplifies to ax + by + cz + d = 0. In vector form: (r - a).n = 0 or r.n = d.

ax + by + cz + d = 0 \quad \text{or} \quad \vec{r} \cdot \hat{n} = p

The plane equation is central to 3D geometry. It appears in distance problems, angle calculations, intersection of planes, and image/foot of perpendicular questions. JEE tests multiple ways of forming the equation: normal form, three-point form, and intercept form.

Plane through three pointsDistance from point to planeAngle between planesIntersection of planes

Practice (11 Qs) →

Foot of Perpendicular and Image of a Point

STANDARD

The foot of the perpendicular from a point P to a plane is found by writing the line through P along the normal to the plane and finding its intersection with the plane. The image of P is twice as far as the foot from P along the same normal direction.

\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} = -\frac{2(ax_1+by_1+cz_1+d)}{a^2+b^2+c^2}

Foot of perpendicular and image problems are popular in JEE because they combine line equations, plane equations, and distance formulas. The image formula (factor of 2) is a common source of errors.

Image of point in planeFoot of perpendicularReflection problemsMirror image

Practice (8 Qs) →

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