Question

The equation to the straight line passing through the point $(a{\cos }^3\theta ,a{\sin }^3\theta )$ and perpendicular to the line $xsec\theta +y\cos ec\theta =a$, is

• A. $x\cos \theta -y\sin \theta =a\cos 2\theta$
• B. $x\cos \theta +y\sin \theta =a\cos 2\theta$
• C. $x\sin \theta +y\cos \theta =a\cos 2\theta$
• D. None of these

Answer 1

The correct option is A

$x\cos \theta -y\sin \theta =a\cos 2\theta$

Explanation for the correct answer:

Step 1: Find the slope of the given line.

A equation of line $xsec\theta +y\cos ec\theta =a$ is given.

Rewrite the equation as follows:

$$\begin{gathered} y=-x\frac{sec\theta }{\cos ec\theta }+\frac{a}{\cos ec\theta } \\ \Rightarrow y=-x\frac{{\displaystyle \frac{1}{\cos \theta }}}{{\displaystyle \frac{1}{\sin \theta }}}+\frac{a}{{\displaystyle \frac{1}{\sin \theta }}} \\ \Rightarrow y=-x\frac{\sin \theta }{\cos \theta }+a\sin \theta ...1 \end{gathered}$$

Since, the general equation of line is:

$y=mx+c...2$

Where, $(x,y)$ is the general point on the line.

$m$ is the slope of the line.

$c$ is the $y-$intercept.

On comparing equation $1$ and equation $2$. we get,

$m=-\frac{\sin \theta }{\cos \theta }$

Therefore, the slope of the line $xsec\theta +y\cos ec\theta =a$ is $-\frac{\sin \theta }{\cos \theta }$.

Step 2: Find the equation of the required line.

We know that, the slope of perpendicular line are negative inverse of each other.

Therefore, the slope $m_1$ of the required line can be given by:

$$\begin{gathered} m_1=-\left(\frac{1}{-{\displaystyle \frac{\sin \theta }{\cos \theta }}}\right) \\ \Rightarrow m_1=\frac{\cos \theta }{\sin \theta } \end{gathered}$$

Also, the line passes through the point $(a{\cos }^3\theta ,a{\sin }^3\theta )$.

We know that, one point slope form of line is:

$y-y_1=m(x-x_1)$

Where, $(x,y)$ is the general point on the line.

$m$ is the slope of the line.

$(x_1,y_1)$ is the given/known point on the line.

Therefore, the equation of the requried line can be given by:

$$\begin{gathered} (y-a{\sin }^3\theta )=\frac{\cos \theta }{\sin \theta }(x-a{\cos }^3\theta ) \\ \Rightarrow y\sin \theta -a{\sin }^4\theta =x\cos \theta -a\theta {\cos }^4\theta \\ \Rightarrow x\cos \theta -y\sin \theta =a{\cos }^4\theta -a{\sin }^4\theta \\ \Rightarrow x\cos \theta -y\sin \theta =a\left({\cos }^4\theta -{\sin }^4\theta \right) \\ \Rightarrow x\cos \theta -y\sin \theta =a\left({\cos }^2\theta -{\sin }^2\theta \right)\left({\cos }^2\theta +{\sin }^2\theta \right) \\ \Rightarrow x\cos \theta -y\sin \theta =a\cos 2\theta \end{gathered}$$

Therefore, the equation of the line which passes through the point $(a{\cos }^3\theta ,a{\sin }^3\theta )$ and perpendicular to the line $xsec\theta +y\cos ec\theta =a$ is $x\cos \theta -y\sin \theta =a\cos 2\theta$.

Hence, option $A$ is the correct .