Question

If $\theta $ is the angle of intersection of curves $y = [|\sin x| + |\cos x|]$ and ${x^2} + {y^2} = 5$. Then the value of $|\tan \theta |$ is

Answer 1

Hint: We will first find the value of y. After finding the value of y we will find the intersection points of the given curves. After it, we will find the value of $|\tan \theta |$.

Complete step-by-step answer:
We are given $y = [|\sin x| + |\cos x|]$. Now, the value $(|\sin x| + |\cos x|)$ $ \in \left[ {1,\sqrt 2 } \right]$. Also, [ ] represents the greatest integer function. So, y = 1.
Now, in the curve ${x^2} + {y^2} = 5$, putting y = 1, as both the curves intersect each other. So, after applying value we get,
${x^2} + {1^2} = 5$
$x = \pm 2$
So, the point of intersection is $( \pm 2,1)$. Now, to find the angle of intersection of the given curves we have to first find the slope of each curve. Curve y = 1 has a slope 0. For finding the slope of the curve ${x^2} + {y^2} = 5$ we have to differentiate the whole curve with respect to x.
Differentiating both sides with respect to x, we get
$2x + 2y\dfrac{{dy}}{{dx}} = 0$
$\dfrac{{dy}}{{dx}} = - \dfrac{x}{y}$
Therefore, the slope of the curve at the point of intersection is 2.
Now, the angle between two curves is found by the formula,
$\tan \theta = \left| {\dfrac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}} \right|$ , where $\theta $ is the angle of intersection between the curves having slopes m1 and m2 respectively.
So, applying the values in the above formula, we get
$\tan \theta = \left| {\dfrac{{0 - ( - 2)}}{{1 - 0}}} \right|$ = $2$
$\tan \theta = 2$
Therefore, $|\tan \theta |$ = 2.

Note: Whenever we come up with such types of questions, we will first find the point of intersection of the given curves. For finding the angle of intersection, we have to find the slope of each curve given. After it, we will use the formula $\tan \theta = \left| {\dfrac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}} \right|$ to find the solution of the given question.