Question

Angle subtended by the common tangents of two ellipses $4(x-4{)}^2+25y^2=100$ and $4(x+1{)}^2+y^2=4$ at origin is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{2}$
• D. None of these

Answer 1

The correct option is C $\frac{\pi }{2}$

Equation of ellipse, $4(x-4)²+25y²=100$

or, $(x-4)²/5²+y²/2²=1$ , it similar of standard equation of ellipse.

and again, equation of ellipse, $4(x+1)²+y²=4$

or, $(x+1)²/1+y²/2²=1$

now, draw both the equation of ellipse ,

you get, a comment tangent as shown in figure . tangent touches first ellipse at $(4,2)$ and 2nd ellipse at $(-1,2),itsubtendsangleatorigin$(0,0)$ .

so, slope of line joining O to 1st ellipse is $1/2$

And slope of line joining O to 2nd ellipse is -2

now, angle made $={\tan }^{-}11/2+2/1+1/2\times -2$

$={\tan }^{-}1\left(\infty \right)=\pi /2$

hence answer is $\pi /2$