Question

pls answer fast guys.....


35 points​

Answer 1

Answer:

(x₁² - ay₁) x + (y₁² - ax₁) y = ax₁y₁

Step-by-step explanation:

Given--->

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Equation of curve

x³ + y³ = 3axy and point on curve is (x₁,y₁)

To prove --->

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Eqyation of tangent at point ( x₁, y₁ ) is

(x₁² - ay₁)x + (y₁² - ax₁)y = ax₁y₁

Proof--->

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x³ + y³ = 3a x y -----------------(1)

Point (x₁ , y₁) is on the curve so it satisfy the equation of curve ie equation (1)

x₁³ + y₁³ = 3a x₁ y₁

x₁³ + y₁³ = 2ax₁ y₁ + a x₁ y₁

x₁³ + y₁³ - 2ax₁y₁ = ax₁y₁

Now differentiating equation (1) with respect to x

d / dx ( x³ + y³ ) = 3a d / dx ( x y )

3 x² + 3 y² dy /dx = 3a ( x dy/dx + y × 1)

x² + y² dy/dx = ax dy /dx + ay

( y² - ax ) dy/dx = ( ay - x² )

(ay - x² )

dy / dx = ---------------

( y² - ax )

ay₁ - x₁²

Slope of tangent at (x₁,y₁)=-------------

y₁² - ax₁

Equation of tangent at (x₁,y₁)

(ay₁ - x₁²)

(y- y₁) = ----------------- (x - x₁)

( y₁² - ax₁)

(y-y₁) ( y₁² - ax₁) = (ay₁ - x₁²) ( x - x₁)

(y₁² - ax₁)y - y₁ (y₁² - ax₁) = (ay₁ -x₁² )x -

(ay₁ - x₁²)x₁

-(ay₁ - x₁²)x + (y₁² - ax₁)y =y₁³-ax₁y₁-ax₁y₁ +x₁³

(x₁² -ay₁)x + (y₁² - ax₁)y = (x₁³ -2ax₁y₁ + y₁³)

putting x₁³ -2ax₁y₁ + y₁³ = ax₁y₁

(x₁² - ay₁)x + (y₁² - ax₁) y = ax₁ y₁

Answer 2

Step-by-step explanation:

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