Find normal line equation at given point

\includegraphics{figure_8} The diagram shows the curve with equation \(y = 2 + 3x - x^2\) and the straight lines \(l\) and \(m\). The line \(l\) is the tangent to the curve at the point \(A\) where the curve crosses the \(y\)-axis.

1. Find an equation for \(l\). [5]

The line \(m\) is the normal to the curve at the point \(B\). Given that \(l\) and \(m\) are parallel,

1. find the coordinates of \(B\). [6]

Find the equation of the normal to the curve \(y = 8x^4 + 4\) at the point where \(x = \frac{1}{2}\). [5]

The curve \(C\) has the equation \(y = x^2 - 5x + 2\ln \frac{x}{3}\), \(x > 0\).

1. Show that the normal to \(C\) at the point where \(x = 3\) has the equation $$3x + 5y + 21 = 0.$$ [5]
2. Find the \(x\)-coordinates of the stationary points of \(C\). [3]

A curve cuts the \(x\)-axis at \((2, 0)\) and has gradient function $$\frac{dy}{dx} = \frac{24}{x^3}$$

1. Find the equation of the curve. [4 marks]
2. Show that the perpendicular bisector of the line joining \(A(-2, 8)\) to \(B(-6, -4)\) is the normal to the curve at \((2, 0)\) [6 marks]

At a point \(P\) on a curve, the gradient of the tangent to the curve is 10 State the gradient of the normal to the curve at \(P\) Circle your answer. [1 mark] \(-10\) \quad \(-0.1\) \quad \(0.1\) \quad \(10\)

A curve with equation \(y = f(x)\) passes through the point \((3, 7)\) Given that \(f'(3) = 0\) find the equation of the normal to the curve at \((3, 7)\) Circle your answer. [1 mark] \(y = \frac{7}{3}x\) \(y = 0\) \(x = 3\) \(x = 7\)

Find the equation of the normal to the curve \(y = 4 \ln(2x - 3)\) at the point where the curve crosses the \(x\) axis. Give your answer in the form \(ax + by + k = 0\) where \(a > 0\). [5]

In this question you must show detailed reasoning Find the equation of the normal to the curve \(y = \frac{x^2-32}{\sqrt{x}}\) at the point on the curve where \(x = 4\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]

In this question you must show detailed reasoning. Find the equation of the normal to the curve \(y = 4\sqrt{x - 3x + 1}\) at the point on the curve where x = 4. Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [7]

The curve \(C\) has equation $$y = \frac{1}{2}x^3 - 9x^2 + \frac{8}{x} + 30, \quad x > 0$$

1. Find \(\frac{dy}{dx}\). [4]
2. Show that the point \(P(4, -8)\) lies on \(C\). [2]
3. Find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [6]

A curve has equation \(y = kx^{\frac{1}{2}}\) where \(k\) is a constant. The point \(P\) on the curve has \(x\)-coordinate 4. The normal to the curve at \(P\) is parallel to the line \(2x + 3y = 0\) and meets the \(x\)-axis at the point \(Q\). The line \(PQ\) is the radius of a circle centre \(P\). Show that \(k = \frac{1}{2}\). Find the equation of the circle. [10]