#### The curve C has equation, y= 9 - 4x - \frac{8}{x}. The point P on C has x-coordinate equal to 2.

The tangent at P meets the x-axis at A and the normal at P meets the x-axis at B.

a) Find the area of triangle APB.

Anonymous

0

The tangent at P meets the x-axis at A and the normal at P meets the x-axis at B.

a) Find the area of triangle APB.

Krishna

0

Step 1: Calculate the point P(any variable) on the curve

NOTE: Substitute the given coordinate in the curve equation to get the

point

EXAMPLE: y = 9 - 4x - \frac{8}{x}, x-coordinate = 2

So, y = 9 - 4*2 - \frac{8}{2}

y = 9 - 8 - 4 = -3

Step 1: Calculate the slope of the curve

Skill 1: Note down the given curve equation and apply the differentiation on

both sides

NOTE: Multiple both sides of the function by \frac{dy}{dx}

Skill 2: Differentiate both sides of the function with respect

to x (any variable).

EXAMPLE: y = x^3 - 2x^2 - x + 9

[math] \frac{d}{dx} y = \frac{d}{dx}[x^3] - \frac{d}{dx} [2x^2] -\frac{d}{dx} x + \frac{d}{dx} 9][/math]

\frac{dy}{dx} = 3x^2 - 4x - 1

Skill 3: Plug in the (x, y) values to find ( \frac{dy}{dx})

for that point.

Skill 4: To know the slope (gradient), apply the BODMAS rules to simplify

the equation.

Step 2: Find the equation of tangent line and the point

Skill 1: Use the slope of curve and a point on curve to find its y-intercept.

NOTE:

Skill 2: Use the slope of line and the y-intercept of line to find the equation

of the line.

EXAMPLE: Plug the slope m = -2 and the y-intercept b = -8 into the

slope- intercept formula.

y = mx + b

y = -2x + -8

Step 3: Find the equation of the normal line to the tangent

NOTE: i) Repeat the step 2 but change the slope.

ii) Perpendicular lines have slopes that are opposite

reciprocals, like \frac{a}{b} \text{ and } \frac{-b}{a}. The slopes also have a product of -1

Step 4: Skill 3: Substitute the corresponding intercept of above lines, to get required point.

NOTE: i) If the line intersects with the x-axis put y = 0 then simplify for the x-coordinator

ii)If the line intersects with the y-axis put x = 0 then simplify for the y-coordinator

Step 5: Calculate the area of the triangle

Method 1:

Step 1: Locate the coordinates of the endpoints.

EXAMPLE: The given points M (2, 3) and N (-1, 0), (2, -4.)

Therefore, (x_1, y_1) = (2, 3),(x_2, y_2) = (-1, 0) and (x_3, y_3) = (2, -4).

Step 2: Plug the corresponding coordinates into the Area of triangle formula

FORMULA:

Area of the triangle

\frac{1}{2} {x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)}

Step 3: Simplify further

NOTE: Apply the BODMAS rules

Method 2:

Step 1: Calculate the base and height lengths of the triangle

NOTE: Use the distance formula to calculate lengths.

Step 2: Substitute the either values (base and height lengths) in the formula

Area of the right angle triangle = \frac{1}{2} base * height

Step 3: Simplify further

NOTE: Apply the BODMAS rules