find the locus

Question

locus of point of intersection of perpendicular tangents drawn one to each of the circles x² + y² = 8 and x² + y² =12 is

(I) x² + y² = 4

(ii)x² + y² = 20

(iii)x² + y² = 208

(iv)x² + y² = 16

Answer 1

Since the circles are nested in each other, I'm thinking that you're talking about the circle that will have the points where these tangents intersect. 

 

Draw a big circle and a smaller circle on a piece of paper. Circumscribe a box around the bigger one. Draw tangents off of the smaller one that are parallel to the sides of this box. Where these lines intersect, we can see we'll have a circle passing through with a radius a bit bigger than the radius of the original circle.

 

If you pick an opportune point of intersection (it helps if you're doing this on graph paper), it's easy to find the radius of this new circle. If R is the radius of the bigger circle and r is the radius of the smaller circle (the ones you started with), you'll have one of these points occur at (R, r). The distance from the origin to this point is the radius of your new circle, √(R² + r²). 

 

If you're writing the equation of a new circle with it's center at the origin and this radius, that equation will look like

 

x² + y² = R² + r²

 

So (ii) should be your answer.