Question

If the coordinates of a point be given by the equations $x = b\sec\varphi,\mspace{6mu}\mspace{6mu} y = a\tan\varphi$, then its locus is.

• A. A straight line
• B. A circle
• C. An ellipse
• D. A hyperbola

Answer 1

Answer: (D)

A hyperbola

Answer 2

Here $\frac { x } { b } = \sec \phi$ and (o) $\frac { y } { a } = \tan \phi$

Therefore

$\frac { x ^ { 2 } } { b ^ { 2 } } - \frac { y ^ { 2 } } { a ^ { 2 } } = \sec ^ { 2 } \phi - \tan ^ { 2 } \phi \Rightarrow \frac { x ^ { 2 } } { b ^ { 2 } } - \frac { y ^ { 2 } } { a ^ { 2 } } = 1$,

which is obviously a hyperbola.