Maths · Circle, conic sections: A standard form of equations of a circle, the general form of the equation of a circle, its radius and centre
The tangents drawn from the origin to the circle 0 are perpendicular, if
The tangents drawn from the origin to the circle \( x^{2}+y^{2}+2 g x+2 f y+f^{2}= \) 0 are perpendicular, if
- A. \( g=f \)
- B. \( g=2 f \)
- C. \( 2 g=f \)
- D. \( 3 g=f \)
Step-by-step solution
The circle equation is x^2+y^2+2gx+2fy+f^2=0. Center: (-g, -f), radius = √(g^2+f^2-f^2)=|g|. The condition for tangents from a point to a circle to be perpendicular is that the point lies on the director circle: (x+g)^2+(y+f)^2=2g^2. Substituting origin (0,0) gives g^2+f^2=2g^2 → f^2=g^2 → f=±g. Among options, only g=f (option A) matches.
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