Question

If $\frac{x}{\alpha }+\frac{y}{\beta }=1$ touches the circle $x^2+y^2=a^2,$ then point $(\frac{1}{\alpha },\frac{1}{\beta })$ lies on a / an ___________.

• A. straight line
• B. circle
• C. parabola
• D. ellipse

Answer 1

Answer: (B)

circle

if line $\frac{x}{\alpha }+\frac{y}{\beta }=1$ touches the circle $x^2+y^2=a^2$,

then distance of the line from the center of the circle should be same as the radius

$\left|\frac{1}{\sqrt{\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}}}\right|=a$

$\Rightarrow \frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}=\frac{1}{a^2}$

Hence the point $(\frac{1}{\alpha },\frac{1}{\beta })$ lie on a circle whose radius is $\frac{1}{a}$ and center at $(0,0)$