Question

Eq. Of the common tangent to the circle x^2+y^2=4 and ellipse 2x^2+y^2=2

Answer 1

Answer:

There are no common tangents!

The points that satisfy x² + y² = 4 are the points of a circle with radius 2 and centred at the origin.

The other equation 2x² + y² = 2 is an ellipse, also centred at the origin, and with axes along the x- and y-axes.  The extremities of the axes are at the points:

(-1 , 0) and (1, 0) for the horizontal axis  [ just put y = 0 and solve for x ]

(0, -√2) and (0, √2) for the vertical axis  [ just put x = 0 and solve for y ].

Since the circle has radius 2, which is greater than both 1 and √2, the ellipse is contained entirely within the circle.  See the attachment.

So any tangent to the ellipse cuts the circle in two points.  There is no common tangent.