Mock Test 2 — Circles

Find the coordinates of the radical center for the system of three circles $x^2+y^2=1$, $x^2+y^2-2x=3$, and $x^2+y^2-4y=5$.

If a variable circle cuts the two fixed orthogonal circles $S_1 = 0$ and $S_2 = 0$ orthogonally, then the locus of its center point forms a:

Find the equation of the direct external common tangent line to the circles $x^2+y^2=4$ and $x^2+y^2-10x+16=0$ if their radii are $r_1 = 2$ and $r_2 = 3$.

The locus of the point of intersection of two mutually perpendicular tangent lines drawn to the circle $x^2 + y^2 = R^2$ forms a concentric circle called the director circle, whose equation matches:

Find the equation of a circle belonging to the coaxial system family $(x^2+y^2-4) + \lambda(x-y) = 0$ that passes through the point $(2, 4)$.

Find the angle of intersection between the two circles $x^2 + y^2 - 4x - 2y + 1 = 0$ and $x^2 + y^2 - 2x - 4y + 1 = 0$.

The length of the transverse internal common tangent line to two separate circles with center distance $d$ and radii $r_1, r_2$ is:

If the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ bisects the circumference of the circle $x^2 + y^2 + 2g'x + 2f'y + c' = 0$, then their common chord line must pass through:

Find the locus of the midpoint of a chord of the circle $x^2 + y^2 = R^2$ that subtends a right angle ($90^\circ$) at the center point.

The numbers of common tangents that can be drawn to the two circles $(x-1)^2 + (y-1)^2 = 1$ and $(x-2)^2 + (y-2)^2 = 4$ is exactly:

If the two circles $x^2 + y^2 = 25$ and $x^2 + y^2 - 24x - 10y + c = 0$ touch each other externally, find the value of the constant parameter $c$ if $r_1=5$ and $r_2=21$.

Find the total number of common tangents that can be drawn to two concentric circles with distinct non-zero radii.

Find the value of the scaling parameter $k$ if the director circle equation for $x^2 + y^2 = 50$ is mapped as $x^2 + y^2 = k$.

Assertion (A): The radical axis of two intersecting circles is identical to their common chord line format equation $S_1 - S_2 = 0$.
Reason (R): The power of any point lying on the common chord line is zero with respect to both circles, satisfying the definitions of the radical axis locus tracker.

Assertion (A): Two circles with equations $x^2 + y^2 - 2x = 0$ and $x^2 + y^2 - 2y = 0$ cut each other orthogonally.
Reason (R): The orthogonality criterion equation evaluates as $2( -1)(0) + 2(0)( -1) = 0 + 0 = 0$, matching the sum of their constant parameters $c_1 + c_2 = 0$.