Circles practice · Mixed difficulty

Option C (-16) is correct because when the area of the circle is $9\pi$, the value of k is -16.

Option B ($\left(\frac{{-a^2 l}}{{n}}, \frac{{-a^2 m}}{{n}}\right)$) is correct because it represents the coordinates of the pole of the given line with respect to the circle, derived from comparing the equation of the polar.

Option A (At most one) is correct because if there were two rational points on the circle, the center would also have to be rational, contradicting the given irrational center.

The correct answer is D (none of these) because the given conditions do not lead to any of the provided options for the range of 'a'.

The correct answer is ${x^2} + {y^2} = \frac{1}{{{{a^2}}}}$ because it represents the condition for the line to be tangent to the circle.

Option D ($x^2+y^2-a x-b y=0$) is correct because it represents the equation of a circle with the given center and radius.

Option C ($3 \sqrt{3} a^2 / 4$) is correct because it is the area of an equilateral triangle inscribed in a circle of radius a.

Option A ($x^2+y^2-30 x+12 y-27=0$) is correct because it represents the equation of the circle with the normal chord PQ as diameter, calculated using the properties of parabolas and normals.

Option B ($\left(\frac{1}{2},-\sqrt{2}\right)$) is correct because it represents the center of the circle passing through the given points and touching the specified circle.

Option A ($2 x+9 y-30=0$) is correct because it represents the equation of the normal to the given circle at the point (-3,4), derived using the normal equation formula.

Option C ($\frac{\pi}{2}$) is correct because the origin lies on the director circle of the given circle, implying the tangents from the origin are perpendicular.

Option B (50) is correct because the condition for one circle bisecting the circumference of another leads to the equation $l+m=50$.

Option A ($\sqrt{5} y-2 x-4=0$) is correct because it represents the common tangent to both the circle and the hyperbola, satisfying the tangency conditions for both curves.

The correct answer is 40 because it is the square of the length of the tangent from the point (3, -4) to the given circle.

Option A $((-1,4))$ is correct because it represents the interval in which 'a' must lie for the point (a, -a) to be inside the given circle.

Option B ($120(x^2+y^2)-22(15 y+8 x)+289=0$) is correct because the equation of the circle with the given diameter is found by using the roots of the two quadratic equations.

Option B ($52\,{r^2}$) is correct because it is the value of ${k^2}$ when the line is tangent to the circle.

Option A (0) is correct because the point (2,2) lies inside the circle, and thus no real tangents can be drawn.

Option D (${x^2} + {y^2} + x + y = 0$) is correct because it represents the circle described on the common chord of the given circles as a diameter, which is found by using the equation $S_1 + \lambda(S_1 - S_2) = 0$ and the condition that the center lies on the common chord.

Option B ($x^2+y^2-3 x-5 y+8=0$) is correct because it represents the circumcircle of the triangle formed by the given lines, found by using the general equation of a circle passing through the vertices of the triangle.

Option B (12/5) is correct because it is the difference between the slopes of the tangents calculated using the equation of the pair of tangents from a point to a circle.

Option A (3) is correct because by equating the derivatives and using the given point, we can solve for a and b and then find the value of (a/2 - b).

Option D (4) is correct because there are four circles that can touch all three lines: one incircle and three excircles of the triangle formed by the lines.

Option B ($4 x+3 y+19=0 \& 4 x+3 y-31=0$) is correct because these are the equations of the tangents to the given circle that are parallel to the given line, found by using the distance from the center to the tangent equals the radius.

The correct answer is B (6) because it represents the radius of the larger circle that satisfies the given conditions.

Option A ($8\pi (3 - 2\sqrt 2 )$) is correct because it represents the area of the smaller circle that touches the parabola at (1, 2) and the x-axis, calculated using geometric properties and calculus.

Option A (${x^2} + {y^2} = 2{a^2}$) is correct because the locus of the point of intersection of two perpendicular tangents to a circle is the director circle, which has the equation ${x^2} + {y^2} = 2{a^2}$.

Option C (72) is correct because by substituting the point and using the tangency condition, we can derive the value of 'c' as 72.

Option A (a circle) is correct because the condition for tangency leads to the equation $h^2 + k^2 = \frac{1}{4}$, which represents a circle.

Option B ($x + y = 2\sqrt 2$) is correct because it satisfies the condition of being tangent to the given circle and having equal intercepts on the positive coordinate axes.