Question

Find the locus of the mid-point of the distance between the axes of the variable line $x\cos \alpha + y\sin \alpha = p$, where p is constant.
A.${x^2} + {y^2} = 4{p^2}$
B.$\dfrac{1}{{{x^2}}} + \dfrac{1}{{{y^2}}} = \dfrac{4}{{{p^2}}}$
C.${x^2} + {y^2} = \dfrac{4}{{{p^2}}}$
D. $\dfrac{1}{{{x^2}}} + \dfrac{1}{{{y^2}}} = \dfrac{2}{{{p^2}}}$

Answer 1

Hint: First rewrite the given line in intercept form and obtain the x and y intercepts of the given line. Then suppose the coordinate of the middle point of the line by the x and y intercepts. Now, use the trigonometric formula of sine and cosine to obtain the required answer.

Formula Used:
In the line equation $\dfrac{x}{a} + \dfrac{y}{b} = 1$ the x intercept is $(a,0)$ and the y intercept is $(0,b)$ .
The trigonometric identity is,
${\sin ^2}x + {\cos ^2}x = 1$.

Complete step by step solution:
The given line equation is $x\cos \alpha + y\sin \alpha = p$.
This line equation can be written as,
$\dfrac{x}{{\dfrac{p}{{\cos \alpha }}}} + \dfrac{y}{{\dfrac{p}{{\sin \alpha }}}} = 1$
Therefore, the x intercept is $(\dfrac{p}{{\cos \alpha }},0)$ and the y intercept is $\left( {0,\dfrac{p}{{\sin \alpha }}} \right)$ .
Suppose that the coordinate of the middle point of the given line is $P(h,k)$ .
Therefore,
$h = \dfrac{{\dfrac{p}{{\cos \alpha }} + 0}}{2}$
$\cos \alpha = \dfrac{p}{{2h}}$
And,
$k = \dfrac{{\dfrac{p}{{\sin \alpha }} + 0}}{2}$
$\sin \alpha = \dfrac{p}{{2k}}$
Substitute the values of $\sin \alpha ,\cos \alpha $ in the trigonometric identity ${\sin ^2}\alpha + {\cos ^2}\alpha = 1$to obtain the required locus.
${\left( {\dfrac{p}{{2k}}} \right)^2} + {\left( {\dfrac{p}{{2h}}} \right)^2} = 1$
$\dfrac{{{p^2}}}{{4{h^2}}} + \dfrac{{{p^2}}}{{4{k^2}}} = 1$
$\dfrac{{{p^2}}}{4}\left( {\dfrac{1}{{{h^2}}} + \dfrac{1}{{{k^2}}}} \right) = 1$
$\dfrac{1}{{{h^2}}} + \dfrac{1}{{{k^2}}} = \dfrac{4}{{{p^2}}}$
Therefore, the required locus is $\dfrac{1}{{{x^2}}} + \dfrac{1}{{{y^2}}} = \dfrac{4}{{{p^2}}}$.

Option ‘B’ is correct

Note: We often obtain the distance between the axis and the given line and then want to obtain the required locus, but that is not the correct way to answer this question. First, obtain the intercepts and then the midpoint of the line passing through the intercepts and then eliminate the variable $\alpha $ to obtain the required locus.