Question

The tangents drawn from the origin to the circle `x^2 + y^2 -2rx -2hy +h^2=0` are perpendicular if?

A) `h=r`
B)`h=-r`
C) `r^2+h^2=1`
D) `r^2+h^2=2`

Answer 1

Both (A) and (B).

Explanation:

The center of circle `x^2+y^2-2rx-2hy+h^2=0` or `(x-r)^2+(y-h)^2=r^2` is `(r,h)` and radius is `r`.

As `x`-coordinate of center is `r` and radius too is `r`, one of the tangent is `x=0`, the `y`-axis and as other tangent would be parallel to `x`-axis.

Further, as we are drawing tangent from origin i.e. `(0,0)`, it can only be `x`-axis i.e. `y=0`.

As such, distance from center `(r,h)` to `x=0` i.e. `+-h=r`.

Hence answer could be both (A) and (B).

As an example see the following graph.

graph{((x-5)^2+(y-5)^2-25)((x-5)^2+(y+5)^2-25)=0 [-19.92, 20.08, -9.76, 10.24]}