Find normal equation at point

3. A curve has the equation $$3 x ^ { 2 } - 2 x + x y + y ^ { 2 } - 11 = 0$$ The point \(P\) on the curve has coordinates \(( - 1,3 )\).

1. Show that the normal to the curve at \(P\) has the equation \(y = 2 - x\).
2. Find the coordinates of the point where the normal to the curve at \(P\) meets the curve again.
   3. continued

15 In this question you must show detailed reasoning. The equation of a curve is $$\ln y + x ^ { 3 } y = 8$$ Find the equation of the normal to the curve at the point where \(y = 1\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are constants to be found.

12 The equation of a curve is $$( x + y ) ^ { 2 } = 4 y + 2 x + 8$$ The curve intersects the positive \(x\)-axis at the point \(P\).
12

1. Show that the gradient of the curve at \(P\) is \(- \frac { 3 } { 2 }\)

   12	Find the equation of the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
   	[2 marks]
   	\(\_\_\_\_\)

5 The curve \(C\) has equation \(x ^ { 2 } + x y + y ^ { 2 } = 19\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 x - y } { x + 2 y }\).
2. Hence find the equation of the normal to \(C\) at the point \(( 2,3 )\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

8 The curve \(C\) has equation \(y ^ { 3 } + 6 y ^ { 2 } - 2 y = 3 x ^ { 2 } + 2 x\). Show that the equation of the normal to \(C\) at the point \(( 1,1 )\) can be written in the form \(8 y + 13 x - 21 = 0\).

The equation of a curve is \(3x^2 + 4xy + y^2 = 24\). Find the equation of the normal to the curve at the point \((1, 3)\), giving your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [8]

The curve \(C\) is defined by the equation $$8x^3 - 3y^2 + 2xy = 9$$ Find an equation of the normal to \(C\) at the point \((2, 5)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [7]

A curve has equation $$x^3 - 2xy - 4x + y^3 - 51 = 0.$$ Find an equation of the normal to the curve at the point \((4, 3)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [8]

The curve \(C\) has equation \(5x^2 + 2xy - 3y^2 + 3 = 0\). The point \(P\) on the curve \(C\) has coordinates \((1, 2)\).

1. Find the gradient of the curve at \(P\). [5]
2. Find the equation of the normal to the curve \(C\) at \(P\), in the form \(y = ax + b\), where \(a\) and \(b\) are constants. [3]

A curve has the equation $$x^2 + 3xy - 2y^2 + 17 = 0.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5]
2. Find an equation for the normal to the curve at the point \((3, -2)\). [3]

A curve has the equation $$3x^2 + xy - 2y^2 + 25 = 0.$$ Find an equation for the normal to the curve at the point with coordinates \((1, 4)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [8]

A curve has the equation $$x^2 + 3xy - 2y^2 + 17 = 0.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [4]
2. Find an equation for the normal to the curve at the point \((3, -2)\). [3]

A curve has equation \(x^3 - 3x^2y + y^2 + 1 = 0\).

1. Show that \(\frac{dy}{dx} = \frac{6xy - 3x^2}{2y - 3x^2}\). [4]
2. Find the equation of the normal to the curve at the point \((1, 2)\). [4]