Question

The power of the point (0, 0) with respect to the circle \[{x^2} + {y^2} + 2x + 4y + 25 = 0\] is
A. 25
B. 24
C. 36
D. 38

Answer 1

Hint: In this particular problem to recall the formula that the power of any point with respect to any conic is equal to the value of that conic at that particular point.

Complete step by step answer:
As we know that the power of any point with respect to any conic is negative when the point is inside that conic and positive when the point is outside that conic. And the power of point is equal to zero when the point lies on the conic.
And to find the power of a point with respect to any conic (here circle) we had to put the coordinates of that point in the equation of conic.
So, here to find the power of point (0, 0) with respect to the given circle \[{x^2} + {y^2} + 2x + 4y + 25 = 0\]. We had to substitute the value of x = 0 and y = 0 in the equation of the circle.
So, the power of point (0, 0) will be \[{\left( 0 \right)^2} + {\left( 0 \right)^2} + 2\left( 0 \right) + 4\left( 0 \right) + 25 = 25\]
Hence, the power of the origin (i.e. (0, 0) with respect to the circle \[{x^2} + {y^2} + 2x + 4y + 25 = 0\] will be 25.

Note: Whenever we face such types of problems then first, we have to recall the definition of the power of a point. And then put the coordinates of the given point in the equation of the circle to find the power of that point. And note that if the position of any point is also asked with respect to any conic then we first find the power of that point and if the power is positive then the position of the point is outside the conic, if the power is negative then the position of that point is inside the conic and if the power is equal to zero then the point lies on the conic.