Each question contains Statement- 1 and Statement- 2 and has...

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct. (a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1 (b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1 (c) Statement- 1 is True, Statement- 2 is False (d) Statement- 1 is False, Statement- 2 is True Statement 1 A common tangent to the circles $x^{2}+y^{2}-6 x=0$ and $x^{2}+y^{2}-10 x+9=0$ is given by $2 y=x+3$ and Statement 2 If two circles touch each other, their radical axis is the common tangent.

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1 A common tangent to the circles $x^{2}+y^{2}-6 x=0$ and $x^{2}+y^{2}-10 x+9=0$ is given by $2 y=x+3$
and
Statement 2
If two circles touch each other, their radical axis is the common tangent.

Key Concepts

Tangent to a Circle

A tangent to a circle is a line that touches the circle at exactly one point, known as the point of tangency. The defining property is that the tangent is perpendicular to the radius drawn to the point of tangency, which provides a geometric condition (the distance from the center of the circle to the line equals the radius) that is frequently used to determine tangency.

Common Tangent

A common tangent is a line that is tangent to two or more circles simultaneously. To find a common tangent, one must ensure that the line satisfies the tangency condition for each circle, meaning that the perpendicular distances from the centers of the circles to the line must equal their respective radii. This concept is central in problems involving multiple circles and their interactions.

Radical Axis

The radical axis of two circles is the locus of all points that have equal power with respect to both circles. It can be determined by subtracting the equations of the two circles, and it often plays a crucial role in identifying special relationships between circles, such as when they touch or intersect. In the specific case of touching circles, the radical axis coincides with the common tangent at the point of tangency.

Power of a Point

The power of a point with respect to a circle is defined as the squared distance from the point to the circle's center minus the square of the circle's radius. This concept is used to compare the relative positions of a point to a circle and underlies the idea of the radical axis, where points with equal power relative to two circles are located.

Transcript

Please give Ace some feedback