Circles practice · Medium

Option D ($\left( { - \frac{1}{2},\frac{1}{2}} \right)$) is correct because the center of the smallest circle lies on the perpendicular bisector of the line segment joining the origin and the intersection of the line y = x + 1 with the perpendicular bisector.

Option A ($\lambda \in(25,32)$) is correct because it satisfies both conditions: the point (2,4) lies inside the circle, and the circle does not cut the axes.

Option B $\left( {\frac{{ - 19}}{{13}},\frac{9}{{13}}} \right)$ is correct because it represents the point of contact when two circles touch each other externally, calculated using the section formula.

Option C ($3x + 4y = \pm \sqrt 5$) is correct because these are the equations of the tangents to the circle that are parallel to the given line.

Option A ($x^2+y^2-4 x+4 y-10=0$) is correct because the circumcircle passes through the vertex and has its center at the centroid.

Option B (2) is correct because there are two integral values of 'a' that satisfy the given conditions for the point to lie inside the larger segment.

Option C ($x^2+y^2-3 x+y-10=0$) is correct because it is derived using the diameter form of the circle equation with the given endpoints.

Option B $((-\infty,-13 / 8) \cup(13 / 8, \infty))$ is correct because it represents the range of lambda values for which the circles have two common tangents.

The correct answer is D (${x^{ - 2}} + {y^{ - 2}} = 4{c^{ - 2}}$) because it represents the locus of the center of the circle passing through O, A, and B, given the line is at a constant distance c from the origin.

The correct answer is A (a circle with radius $\frac{8}{3}$) because the given equation simplifies to the equation of a circle, and the radius is calculated to be $\frac{8}{3}$.

Option B ($y - axis$) is correct because the radius of the circle is equal to the x-coordinate of the center, indicating that the circle touches the y-axis.

Option C ($x+y+1=0$) is correct. The tangent to the parabola at the point where the distance to the circle's center is minimized is found by using the normal to the parabola.

Option B (circle) is correct because the locus of the point of intersection of tangents is a circle.

Option D (None of these) is correct because the points of contact are calculated using the formula involving the slope and radius, and the result doesn't match any of the given options.

Option B ($- 35 < m < 15$) is correct because the line intersects the circle at two distinct points if the distance of the center from the line is less than the radius.

Option A ((1,1)) is correct because the circle passing through the distinct points (1, t), (t, 1), and (t, t) for all values of t, passes through the point (1,1).

Option A ($x^2+y^2-a x-b y=0$) is correct because it represents the circle passing through the origin and the intercepts of the given line.

Option D (Data are inconsistent) is correct because the given conditions lead to conflicting values for the constant term 'c' in the circle's equation.

Option A ($|xy| < \frac{{1}}{{8}}$) is correct because it is derived from the given condition $a \leq \sin A$ and applying AM-GM inequality inside the circumcircle.

Option D (3) is correct because the distance between the centers equals the sum of the radii, indicating the circles touch externally, allowing for three common tangents.

Option B (${\alpha ^2} + {\beta ^2} - {r^2}$) is correct because it represents the power of the point P with respect to the circle.

Option D ($x^2+y^2=72$) is correct because the locus of the point from which mutually perpendicular tangents can be drawn to a circle is the director circle, and its radius squared is twice the radius squared of the original circle.

Option C (4x + 3y + 19 = 0 and 4x + 3y - 31 = 0) is correct because these are the tangent lines found by using the perpendicular distance from the center of the circle to the tangent equals the radius.

Option A is correct because if (α,β) is a point on the chord PQ, then α must lie between the x-coordinates of P and Q, and β must lie between the y-coordinates of P and Q.

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 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 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 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.