Question

Find the angle of intersection of the curves:

[ a) , y = 6 - x ]
[ b) , x^2 = 2y ]

a. (30∘)
b. (45∘)
c. (60∘)
d. (90∘)

Answer 1

The angle of intersection between the curves y = 6 - x and x² = 2y is
`\( (60^\circ) \)`, corresponding to option (c).

Step-by-step explanation:

To find the angle of intersection between the curves, we need to determine the slopes of the two curves at the point of intersection. First, find the point of intersection by solving the system of equations:

y = 6 - x

x²= 2y

Substitute the expression for y from the first equation into the second equation:

x² = 2(6 - x)

Solving this quadratic equation gives the x-coordinate of the point of intersection. Substituting this value back into either equation will provide the y-coordinate.

Next, find the slopes of the curves at the point of intersection. The slopes are given by the derivatives of the respective equations. For y = 6 - x, the slope is -1. For x² = 2y, the slope can be found by differentiating implicitly with respect to x.

Once the slopes are determined, use the formula for the angle between two lines:

`\[ \tan(\theta) = \frac{{m_2 - m_1}}{{1 + m_1 \cdot m_2}} \]`

where m₁ and m₂ are the slopes of the two curves. Finally, find the angle θ by taking the arctangent of the calculated value.