Circles practice · Mixed difficulty

Option D (A circle of radius zero) is correct because the radius of the circle is zero when ${\mathrm{g}^2} + {f^2} = c$.

Option D ($\pi$) is correct because the area of the circle is calculated using the radius, which is half the distance between the two parallel lines.

Option B ($x^2+y^2-2 x-4 y=0$) is correct because it represents the circle passing through (2,0) and (0,4) with the minimum radius, which is the circle with the line segment joining the points as the diameter.

Option C ($x=2$) is correct because the equation of the common tangent is found by subtracting the equations of the two circles, which touch each other externally.

Option B (Circle) is correct because the locus of the point (1/α, 1/β) is a circle, derived from the condition of tangency to the given circle.

Option C ($\sqrt{3} y=x+3$) is correct because it represents the common tangent to the given circle and parabola above the x-axis, found by equating the perpendicular distance from the circle's center to the tangent with the circle's radius.

Option B ($c=35$) is correct because the condition for a point to lie inside and outside the circle gives a range for c, and 35 is the only value that satisfies both conditions.

Option B (${x^2} + {y^2} = 9$) is correct because it represents the locus of the centroid of the triangle formed by the rod and the axes.

Option C ($\theta \in\left(n \pi-\frac{\pi}{6}, n \pi+\frac{\pi}{6}\right) ; n \in Z$) is correct because the condition for non-real roots of the quadratic equation derived from the midpoint of the chord leads to the specified interval for $\theta$.

Option C ($5 x^2+5 y^2+70 y+33=0$) is correct because it represents the locus of a point where the ratio of the lengths of tangents to the given circles is a constant.

Option D is correct because the radius of the circle is calculated to be 0, representing a point circle.

Option B ($\cos ^2 \alpha=1$) is correct because the condition for tangency (distance from center to the line equals the radius) leads to this result.

Option A (externally at (0,1)) is correct because the distance between the centers equals the sum of the radii, and the common tangent equation gives the point of contact.

The correct answer is B ($- 5,5$) because these values of p and q satisfy the conditions for the circle touching the line x=y at the origin and passing through (2,1).

The correct answer is D ($x – y = 2a$) because the center of the circle lies on this line, making it a diameter.

The correct answer is C. For the given equation to represent the circumcircle, it must satisfy the conditions for a circle, which leads to both equations in options A and B holding true.

Option A ($b^2(p^2+q^2)=(a^2-c p-d q)^2$) is correct because it represents the condition where the polar of a point with respect to one circle touches another circle, derived using the distance from the center to the line.

Option C ($\sqrt{2}$) is correct because the radius is calculated using the distance formula between the center of the circle and a point on the circle.

Option A ($3 x+4 y-25=0,4 x-3 y-25=0$) is correct because these are the equations of the tangents derived by finding the slopes of the lines passing through the given point and tangent to the circle.

Option C ($-1 \pm \sqrt{2}$) is correct because it represents the values of m for which the given chord subtends an angle of 45 degrees at the major segment of the circle.

Option A ($2 x+y-7=0$) is correct because it is the equation of the chord bisected at the given point, calculated using T = S'.

Option D ($x-2 y+2=0 \& 2 x+y-1=0$) is correct because it represents the pair of tangents to the given circle from the specified point, derived using the equation $SS_1 = T^2$.

Option B ($\left(\frac{{a}}{{\sqrt{{2}}}}, \frac{{a}}{{\sqrt{{2}}}}\right)$) is correct because the area of the triangle formed by the tangent and the axes is $a^2$, which leads to $\theta = \frac{{\pi}}{{4}}$.

Option A (6) is correct because the sum of a and b in the circle's equation is 6, derived from the chord and circle intersection.

The correct answer is $x^2+y^2-3 \sqrt{2} x-y+4=0$ because it represents the equation of the circle that touches the conic at the given point and passes through one of its foci.

Option B (a circle) is correct because the locus of the centroid forms a circle with a radius related to the original circle's radius.

Option C (GP) is correct because the radii ${r_1},{r_2},{r_3}$ are in geometric progression.

Option A ($x - 3y = 0$) is correct because it represents the other tangent to the circle from the origin, found by using the distance from the center to the tangent.

Option B ($2 \sqrt{2}$) is correct because it is the length of the tangent calculated using the formula $\sqrt{S_1}$.

Option C (${x^2} + {y^2} - 2x - 4y - 3 = 0$) is correct because it represents a circle concentric with the given circle and passing through the point (3,4).