Question

If x sinα + y cosα = p and x cosα – y sinα = q, then (x2 – p2) + (y2 – q2) is

Answer 1

Let the line xcosα+ysinα=p touches the curve at point P(x1,y1)

The equation of tangent at P(x1,y1)

a2xx1−b2yy1=1 ...(1)

Here both the equation represent same line 

Now the equation xcosα+ysinα=p

or αxcosα+ypsinα=1....(2)

Now coefficient of line (1) and (2) are same 

∴a2x1=pcosα   and  b2−y1=psinα

or,  x1=pa2cosα   and   y1=p−b2sinα

PLS MARK BRAINLIEST

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Answer 2

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\:xsin\alpha + ycos\alpha = p - - - (1)$

and

$\rm :\longmapsto\:xcos\alpha - ysin\alpha = q - - - (2)$

Now, Consider

$\red{\rm :\longmapsto\: {p}^{2}+{q}^{2}}$

On substituting the values of p and q, we get

$\rm \: = \: {(xsin\alpha + ycos\alpha )}^{2} + {(xcos\alpha - ysin\alpha )}^{2}$

$\rm \: = \: {x}^{2} {sin}^{2}\alpha + {y}^{2} {cos}^{2}\alpha + 2xysin\alpha \: cos\alpha + \\ \rm \: {x}^{2} {cos}^{2}\alpha + {y}^{2} {sin}^{2}\alpha - 2xy \: sin\alpha \: cos\alpha$

$\rm \: = \: {x}^{2} {sin}^{2}\alpha + {y}^{2} {cos}^{2}\alpha + {x}^{2} {cos}^{2}\alpha + {y}^{2} {sin}^{2}\alpha$

$\rm \: = \: {x}^{2}( {sin}^{2}\alpha + {cos}^{2}\alpha ) + {y}^{2}( {sin}^{2}\alpha + {cos}^{2}\alpha )$

$\rm \: = \: {x}^{2} + {y}^{2}$

$\rm \implies\: {p}^{2} + {q}^{2} = {x}^{2} + {y}^{2}$

$\rm \implies\: {x}^{2} + {y}^{2} - {p}^{2} - {q}^{2} = 0$

$\bf \implies\: ({x}^{2} - {p}^{2}) + ({y}^{2}- {q}^{2})= 0$

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Formula Used :-

$\boxed{ \tt{ \: {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy \: }}$

$\boxed{ \tt{ \: {(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy \: }}$

$\boxed{ \tt{ \: {sin}^{2}x + {cos}^{2}x = 1 \: }}$

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Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1