Question

find the inclination of tangent to the following curve with x axis of given point y^2= 6x +7 at (1,2)​

Answer 1

EXPLANATION.

Point on the curve.

⇒ y² = 6x + 7 at (1,2).

As we know that,

Differentiate the equation w.r.t x, we get.

⇒ 2y.dy/dx = 6 + 0.

⇒ 2y.dy/dx = 6.

⇒ dy/dx = 6/2y.

⇒ dy/dx at a point = (1,2).

Put the value of y = 2 in equation, we get.

⇒ dy/dx = 6/2(2).

⇒ dy/dx = 3/2.

Slope of the equation = 3/2.

As we know that,

Equation of tangent.

⇒ (y - y₁) = m(x - x₁).

⇒ (y - 2) = 3/2(x - 1).

⇒ 2(y - 2) = 3(x - 1).

⇒ 2y - 4 = 3x - 3.

⇒ 2y - 3x - 4 + 3 = 0.

⇒ 2y - 3x - 1 = 0.

                                                                                                                     

MORE INFORMATION.

Point of inflection.

If at any point P, the curve is concave on one side and convex on other sides with respect to x-axes, then the point P is called the point of inflection. Thus P is a point of inflection if at P,

d²y/dx² = 0, but d³y/dx³ ≠ 0.

Also point P is a point of inflection if,

f''(x) = f'''(x) = ,,,,f⁽ⁿ⁻¹⁾ (x) = 0 and fⁿ(x) ≠ 0 for odd n.

Answer 2

Given:-

• Curve ⟹ y² = 6x + 7 at (1 , 2).

To Find:-

• Inclination of Tangent

Criteria used:-

• Equation of Tangent:- $(y-y')=m(x-x')$

Solution:-

As given (1 , 2) are the points. So,

• x' = 1
• y' = 2

Firstly we will find the slope (m) of the equation,

$Slope\:(m)\:⟹\:2y×\dfrac{dy}{dx}=6$

$Slope\:(m)\:⟹\:\dfrac{dy}{dx}=\dfrac{6}{2y}$

Putting the value of y,

$Slope\:(m)\:⟹\:\dfrac{dy}{dx}=\dfrac{6}{2(2)}$

$Slope\:(m)\:⟹\:\dfrac{dy}{dx}=\dfrac{3}{2}$

${\boxed{Slope\:(m)\:=\dfrac{3}{2}}}$

Putting the values in the Equation of Tangent,

$⟹\:(y-y')=m(x-x')$

$⟹\:(y-2)=\dfrac{3}{2}(x-1)$

$⟹\:2(y-2)=3(x-1)$

$⟹\:2y-4=3x-3$

$⟹\:2y-3x=4-3$

$⟹\:2y-3x-1=0$

Hence, the Equation of the Tangent is

2y - 3x - 1 = 0

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