find the locus of the mid point of the portion of the line x cosa +yp sin a = p - Maths - Conic Sections

Question

find the locus of the mid point of the portion of the line x
cosa +yp sin a = p which is intercepted between the axis ??

ramandeep.singh · Accepted Answer

If the equation of the given line is $x \cos α + y \sin α = p$ , then the solution is shown below:

The line $x \cos α + y \sin α = p$ intercepts the axes.

Thus the coordinate of the point where the line intercepts x-axis is $(\frac{p}{\cos α}, 0)$ , and the coordinate of the point where the line intercepts y-axis is $(0, \frac{p}{\sin α})$ . The mid-point R of the line is given by

$R (h, k) = (\frac{\frac{p}{\cos α} + 0}{2}, \frac{0 + \frac{p}{\sin α}}{2}) = (\frac{p}{2 \cos α}, \frac{p}{2 \sin α}) ∴ h = \frac{p}{2 \cos α}, k = \frac{p}{2 \sin α}$

Eliminating the sine and cosine terms, we get

$\cos^{2} α + \sin^{2} α = 1 \Rightarrow \frac{p^{2}}{4 h^{2}} + \frac{p^{2}}{4 k^{2}} = 1 \Rightarrow p^{2} (h^{2} + k^{2}) = 4 h^{2} k^{2}$

Thus, the locus is given by

$p^{2} (x^{2} + y^{2}) = 4 x^{2} y^{2}$

find the locus of the mid point of the portion of the line x cosa +yp sin a = p which is intercepted between the axis ??