Question

write the number of points of intersection of the curves $2y=-1$ and $y=\cos ecx$

Answer 1

Hint: To find the point of intersection algebraically, solve each equation for y, set the two expressions for y equal to each other, solve for x, and plug the value of x into either of the original equations to find the corresponding y-value. The values of x and y are the x and y values of the point of intersection.

Complete step-by-step answer:

The given equations of the curves are

$2y=-1.................(1)$

$y=\cos ecx...............(2)$

From the equation (1), we get

$y=\dfrac{-1}{2}$

Now put this value in the equation (2), we get

$\cos ecx=\dfrac{-1}{2}$

We know that $\cos ecx=\dfrac{1}{\sin x}$

$\dfrac{1}{\sin x}=\dfrac{-1}{2}$

Taking the reciprocal on the both sides, we get

$\sin x=-2$

Since, the solution set of the sine function lies between -1 and 1.

Therefore, the two curves will not intersect at any point.

Hence, the number of points if the intersection of the curves is 0.

Note: The domain of the function $y=\sin x$ is all real numbers (sine is defined for any angle measure), the range is $-1\le y\le 1$. The domain of the function $y=\cos ecx=\dfrac{1}{\sin x}$ is all real numbers except the values where $\sin x$ is equal to 0, that is the values $n\pi $ for all integers n. The range of the function is $y\le -1\text{ or }y\ge 1$.