If x- y are positive integers and tan-1x-cos-1y-1-y2-sin-13-10 then number of ordered pairs of x- y is-

Since y gt; 0 given equation can be re written as tan^ 1x+ tan^ 11/y=tan^ 13 ⇒x+(1/y)/1 (x/y)=3 ⇒ xy+1=3y 3x ⇒ y =3x +1/3 x Since x and y are positive integ ...

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Standard XI

Mathematics

Properties Derived from Trigonometric Identities

If x, y are p...

Question

If x, y are positive integers and $t a n^{- 1} x + c o s^{- 1} \frac{y}{\sqrt{1 + y^{2}}} = s i n^{- 1} \frac{3}{\sqrt{10}}$ then number of ordered pairs of (x, y) is:

infinitely many

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one

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two

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Solution

The correct option is C two
Since y > 0 given equation can be re-written as
$t a n^{- 1} x + t a n^{- 1} \frac{1}{y} = t a n^{- 1} 3 \Rightarrow \frac{x + (\frac{1}{y})}{1 - (\frac{x}{y})} = 3 \Rightarrow x y + 1 = 3 y - 3 x \Rightarrow y = \frac{3 x + 1}{3 - x}$
Since x and y are positive integers,
$x = 1 \Rightarrow y = 2, x = 2 \Rightarrow y = 7$ .
Hence ordered pairs (x, y) = (1, 2), (2, 7)

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Similar questions

Q. If x, y are positive integers and

t a n^{- 1} x + c o s^{- 1} \frac{y}{\sqrt{1 + y^{2}}} = s i n^{- 1} \frac{3}{\sqrt{10}}

then number of ordered pairs of (x, y) is:

Q. The number of integral pairs

(x, y)

with

y > 0

satisfying

{tan}^{- 1} x + {cos}^{- 1} \frac{y}{\sqrt{1 + y^{2}}} = {sin}^{- 1} \frac{3}{\sqrt{10}}

Q. The number of integral pairs

(x, y)

with

y > 0

satisfying

{tan}^{- 1} x + {cos}^{- 1} \frac{y}{\sqrt{1 + y^{2}}} = {sin}^{- 1} \frac{3}{\sqrt{10}}

Q. Inverse circular functions,Principal values of

{s i n}^{- 1} x, {c o s}^{- 1} x, {t a n}^{- 1} x

{t a n}^{- 1} x + {t a n}^{- 1} y = {t a n}^{- 1} \frac{x + y}{1 - x y}

x y < 1

π + {t a n}^{- 1} \frac{x + y}{1 - x y}

x y > 1

.
(a) Solve the equation

{t a n}^{- 1} 2 x + {t a n}^{- 1} 3 x = n π + (π / 4)

.
(b) Find all the positive integral solutions of

{t a n}^{- 1} x + {c o s}^{- 1} (\frac{y}{\sqrt{1 + y^{2}}}) = {s i n}^{- 1} (\frac{3}{\sqrt{10}})

Q. The number of the positive integral solutions of

{tan}^{- 1} x + {cos}^{- 1} (\frac{y}{\sqrt{(1 + y^{2})}}) = {sin}^{- 1} (\frac{3}{\sqrt{10}})

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If x, y are positive integers and tan−1x+cos−1y√1+y2=sin−13√10 then number of ordered pairs of (x, y) is:

The correct option is C twoSince y > 0 given equation can be re-written as tan−1x+tan−11y=tan−13⇒x+(1y)1−(xy)=3⇒xy+1=3y−3x⇒y=3x+13−x Since x and y are positive integers, x=1⇒y=2,x=2⇒y=7. Hence ordered pairs (x, y) = (1, 2), (2, 7)

If x, y are positive integers and $t a n^{- 1} x + c o s^{- 1} \frac{y}{\sqrt{1 + y^{2}}} = s i n^{- 1} \frac{3}{\sqrt{10}}$ then number of ordered pairs of (x, y) is: