Question

P and Q are two point observed from the top of a building 10√3 m hight . if the angleof depression of the point are complementry and PQ = 20m , then the distance of P from the buliding is

SOLUTION​

Answer 1

Step-by-step explanation:

☘Answer:-

✪Solution:-

Angles of depression are x and 90-x.

Tan x = 10√3 / PR

=> PR = 10√3 /Tan x

Tan (90-x) = Cot x = 10√3 /QR

=> QR = 10√3 tan x

PR - QR = 20 m = 10√3 (1/tanx - tan x)

=> Tan² x + (2/√3) tan x - 1 = 0

=> Tan x = 1/√3 by using the solution of quadratic equation.

=> x = 30°

and hence, 90 - x = 60°

Now QR = 10√3 Tan x = 10 m

=> PR = 20 +10 = 30 meters

✿ Hence the distance of P from the building is 30 meter .

Answer 2

▪Question :-

P and Q are two point observed from the top of a building 10√3 m height . If the angle of depression of the point are complementry and PQ = 20m , then the distance of P from the buliding is(Given P is farther point)

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▪Answer :-

Distance of point P from building is 30m.

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▪Solution :-

As the angles are complementary for their sum must be 90°.

According to the Attached Figure:

$\large \bf In \: \triangle \: SRQ$

$\sf \tan(90 - x) = \dfrac{10 \sqrt{3} }{y} \\ \\ \bf \longrightarrow cot \: x = \frac{10 \sqrt{3} }{y} \: \: \: \:. \: . \: . \: (i)$

Now ,

$\bf \large In \: \triangle \: SRP$

$\bf tan \: x = \dfrac{10 \sqrt{3} }{y + 20} \: \: \: \: . \: .\: . \: (ii)$

Mulyiplying (i) and (ii) We get,

$\sf \tan x. \cot x = \dfrac{10 \sqrt{3}}{y} \times \frac{10 \sqrt{3} }{20 + y} \\ \\ \longrightarrow \sf 1 = \frac{300}{y(y + 20)} \\ \\ \longrightarrow \sf1 = \frac{300}{y {}^{2} + 20y} \\ \\ \bf \longrightarrow y {}^{2} + 20y - 300 = 0 \\ \\ \longrightarrow \sf y {}^{2} - 10y + 30y - 300 = 0 \\ \\ \longrightarrow \sf y(y - 10) + 30(y - 10) = 0 \\ \\ \longrightarrow\sf (y - 10)(y + 30) = 0 \\ \\ \large \sf \purple{ \longrightarrow \underline {\boxed{{\bf y = 10 \: \: or \: \: y = - 30 } }}}$

As y is distance So it can't be Negative

Hence,

$\Large\purple{ \longrightarrow \underline {\boxed{{\bf y = 10m} }}}$

So,

Distance of point P from building = 10 + 20

i.e.

$\pink{ \huge \mathfrak{Distance=30m}}$

$\Large \red{\mathfrak{  \text{W}hich \:   \: is  \:  \: the  \:  \: required} }\\ \huge \red{\mathfrak{ \text{ A}nswer.}}$

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