Circles practice · Mixed difficulty

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 A (Assertion is correct, reason is correct; reason is a correct explanation for assertion) is correct because both the assertion about the radius and the reason explaining the radius formula are accurate and related.

Option A (a straight line) is correct because the points P, A, and B are collinear when (PA)² + (PB)² is maximized, resulting in a straight line.

Option B ($14+5 \sqrt{3}$) is correct because it represents the maximum value of (PQ)² calculated by parameterizing the circle and maximizing the distance.

Option C (Outside the circle) is correct because substituting the point P(-1, 0) into the circle's equation results in a positive value, indicating the point lies outside the circle.

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.

Option D (all of these) is correct because the given conditions lead to the equation having a root in [-1, 0] and [2, 5], and distinct roots greater than -1 and less than 5.

Option B ($3{x^2} + 3{y^2} - 13y = 0$) is correct because it is derived from the conditions that the circle's center lies on the y-axis and passes through the origin and the point (2,3).

Option A, $x^2+y^2-6x-8y-25=0$, is the correct locus because it represents the director circle of the given circle, where tangents from any point on it are perpendicular.

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 B ($\frac{{1}}{{a^2}}+\frac{{1}}{{b^2}}=\frac{{1}}{{c^2}}$) is correct because it is derived from the condition that the distance between the centers equals the sum or difference of the radii for touching circles.

Option A ($x-\sqrt{3} y=1$) is correct because it is a tangent to the circle $(x-3)^2+y^2=1$ and is perpendicular to the tangent at P.

Option A (2) is correct because the square of the distance is calculated using the coordinates of point P and the center of circle C1.

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 ($\left( {\frac{{{x_1} + {x_2}}}{2},\frac{{{y_1} + {y_2}}}{2}} \right)$) is correct because the center of the circle is the midpoint of the diameter, and the given equation represents a circle with endpoints of a diameter at $(x_1, y_1)$ and $(x_2, y_2)$.

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 (a circle) is correct because the condition for tangency leads to the equation $\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{1}{a^2}$, which represents a circle.

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 (4) is correct because the intersection of a circle and an ellipse can have a maximum of 4 points.

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 B ($\alpha^2+\beta^2-r^2$) is correct because it represents the power of the point P with respect to the circle, which is equal to PA.PB.

Option D (no value of $\theta$) is correct because the condition for the line to touch the circle leads to an impossible value for $\cos \theta$.

The correct answer is $l \in(-8,8)$ because the length of the perpendicular from the center to the chord must be less than the radius.

Option A $\left( {3,\frac{3}{2}} \right)$ is correct because the center of the circle is the intersection point of its normals.

Option A $\left( {3,\frac{3}{2}} \right)$ is correct because the normals of a circle pass through its center, and the intersection of the given normals gives the center.

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 A ($a^2+b^2$) is correct because the product of the distances CQ and CR simplifies to $a^2 + b^2$ using the equation of the tangent to the hyperbola.