Answer

Verified

447k+ views

**Hint:**Hint! Let the foot of perpendicular be p(a, b, c). Then directions ratio of OP are (a − 0, b − 0, c − 0) = (a, b c) then, direction ratios of the normal to a, b, c is

a(x − a) + b(y − b) + c(z − c) = 0

**Complete step by step solution:**Given that,

the foot of the perpendicular from the origin to a plane is (a, b, c).

Now,

According to the question.

⇒ Let the foot of the perpendicular be P(a, b, c).

Then,

⇒ direction ratios of OP are (a − 0, b − 0, c − 0) = (a, b, c).

So,

⇒ the equation of plane passing through P(a, b, c)

then,

the direction ratios of the normal to which are a, b, c is

$\text{d}\cdot {{\text{r}}_{8}}(x-{{x}_{1}})+\text{d}\cdot \text{r}(\text{y}-{{\text{y}}_{1}})+\text{d}\cdot {{\text{r}}_{8}}(\text{z}-{{\text{z}}_{1}})=0$

where,

${{x}_{1}}$ = x-co-ordinates of point P.

${{\text{y}}_{1}}$ = y-co-ordinates of point P.

${{\text{z}}_{1}}$ = z-co-ordinates of point P.

Now,

⇒ Here, \[{{x}_{1}}=\text{a}\], \[{{\text{y}}_{1}}=\text{b}\], \[{{\text{z}}_{1}}=\text{c}\].

So,

⇒ a(x − a) + b(y − b) + c(z − c) = 0

⇒ After multiplying, we get;

⇒ $\text{a}x-{{\text{a}}^{2}}+\text{by}-{{\text{b}}^{2}}+\text{cz}-{{\text{c}}^{2}}=0$

⇒ ax + by + cz = ${{\text{a}}^{2}}+{{\text{b}}^{2}}+{{\text{c}}^{2}}$.

Hence, the correct option is (C).

**Note:**In this type of question we know about the direction ratios that is If a, b, c are three numbers proportional to the direction cosine l, m, n of a straight line, then a, b, c are called its direction ratios..then, First, find out the direction ratios of line and then find direction ratios of the normal to point (a, b, c)

= $\text{d}\cdot {{\text{r}}_{8}}(x-{{x}_{1}})+\text{d}\cdot {{\text{r}}_{8}}(\text{y}-{{\text{y}}_{1}})+\text{d}\cdot {{\text{r}}_{8}}(\text{z}-{{\text{z}}_{1}})=0$

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which are the Top 10 Largest Countries of the World?

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Write a letter to the principal requesting him to grant class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE