find the diameter

Question

the locus of the point of intersection of perpendicular tangents drawn to each of circles x² + y² = 16 and x² + y² = 9 is a circle whose diameter is

(I)5

(ii)√7

(iii)2√7

(iv)10

Answer 1

Draw the circles on graph paper.

 

Work with vertical and horizontal tangents since they're easy to draw and you only need a few to see the pattern.

 

 

The vertical line x = 3 is tangent to the smaller circle.  The vertical line x = 4 is tangent to the larger circle.

 

The horizontal lines y = 3 and y = 4 are tangent to the small and large circle, respectively.

 

Although those four lines make four intersections (3, 3), (3, 4), (4, 3) and (4, 4), you're only interested in intersections between a tangent of the small circle and a tangent of the big circle.  Those would be (3, 4) and (4, 3).

 

The corresponding intersections in the fourth quadrant (formed by the horizontal lines y = - 3 and y = - 4) are (4, - 3) and (3, - 4).

 

In the second quadrant, you have (- 3, 4) and (- 4, 3).

 

All of these points are on the same circle, so if you can connect any pair of them with a segment passing through the origin, that segment is a diameter.

 

(- 3, 4) and (3, - 4) are one such pair.  With another point in the locus, (- 3, - 4), you have a (3, 4, 5) right triangle, whose hypotenuse is 10.  This is also illustrates the theorem that if one side of an inscribed triangle is a diameter, then the triangle is a right triangle.