Question

Calculate the area of the triangle formed by the tangent to the graph of the function f(x) = (x-6)/(x-2) at the point x = 3 with the axes of the coordinate system.

Answer 1

The area of the triangle formed by the tangent to the graph of the function f(x) = (x-6)/(x-2) at the point x = 3 with the axes of the coordinate system is 28.125 square units.

Explanation:

Differentiation is an algebraic process that finds the gradient (slope) of a curve. At a point, the gradient of a curve is the same as the gradient of the tangent line to the curve at that point.

Given function:

`f(x)=(x-6)/(x-2)`

Differentiate the given function using the quotient rule.

`\boxed{\begin{minipage}{5.5 cm}\underline{Quotient Rule for Differentiation}\\\\If $y=(u)/(v)$ then:\\\\$\frac{\text{d}y}{\text{d}x}=\frac{v \frac{\text{d}u}{\text{d}x}-u\frac{\text{d}v}{\text{d}x}}{v^2}$\\\end{minipage}}`

`\implies f'(x)=((x-2)\cdot 1-(x-6)\cdot 1)/((x-2)^2)`

`\implies f'(x)=(4)/((x-2)^2)`

To find the gradient of the tangent lines at x = 3, substitute x = 3 into the differentiated function:

`\implies f'(3)=(4)/((3-2)^2)=4`

Substitute x = 3 into the function to find the y-value of the point on the curve when x = 3:

`\implies f(3)=(3-6)/(3-2)=-3`

The slope-intercept form of a linear equation is y = mx + b, where m is the gradient and b is the y-intercept.

Substitute the point (3, -3) and the found gradient m = 4 into the slope-intercept formula and solve for b:

`\begin{aligned}y&=mx+b\\\implies-3&=4(3)+b\\-3&=12+b\\b&=-15\end{aligned}`

Therefore, the equation of the tangent to the curve at point x = 3 is:

• 
  `y=4x-15`

To calculate the point at which the tangent line intersects the x-axis, substitute y = 0 into the equation of the tangent:

`\begin{aligned}\implies 0&=4x-15\\4x&=15\\x&=3.75\end{aligned}`

To calculate the point at which the tangent line intersects the y-axis, substitute x = 0 into the equation of the tangent:

`\implies y=4(0)-15=-15`

Therefore, the tangent line intersects the x-axis at (3.75, 0) and the y-axis at (0, -15).

This means the triangle formed by the tangent and the axes of the coordinate system has a height of 15 units and a base of 3.75 units.

`\begin{aligned}\textsf{Area of a triangle}&=(1)/(2)\sf \cdot base \cdot height\\\\\implies \sf Area&=(1)/(2) \cdot 3.75 \cdot 15\\\\&=(225)/(8)\\\\&=28.125\;\; \sf square\;units\end{aligned}`

Therefore, the area of the triangle is 28.125 square units.