Circles practice · Easy

Option A ((9,3)) is correct because it is the only point that lies on the circle and is farthest from the origin.

Option D ($\frac{{39}}{4}$) is correct because it is the value of k that satisfies the given ratio of tangent lengths from the point (1,2) to the two circles.

Option C ($\pm 4 \sqrt{2}$) is correct because it is the value of b for which the common chord subtends a right angle at the origin.

Option C ($(a, b), c$) is correct because the given parametric equations represent a circle with center (a, b) and radius c.

Option B (($-7, 8$)) is correct because it represents the center of the circle formed by the locus of point P, derived from the given ratio of tangent lengths.

Option D ($(2, - 2)$) is correct because it lies on the polar of (1, -1) with respect to the circle ${x^2} + {y^2} = 4$.

Option C ($x \cos \alpha+y \sin \alpha=a(1+\cos \alpha)$) is correct because it represents the equation of the tangent to the given circle at the specified point, derived using the tangent equation formula.

Option A (12) is correct because it is the value of (a+b-c) after finding the equation of the circle using the diameter form and comparing it with the given equation.

Option A ($y^2-x^2=d^2$) is the correct locus because it is derived from the geometric conditions that the circle touches one axis and cuts a constant length from the other.

The correct answer is B (${x^2} + {y^2} - 2hx - 2ky + {h^2} = 0$) because it represents the equation of a circle with center (h, k) touching the x-axis.

Option A, 7, is correct because solving the equations derived from the tangent and normal conditions gives $\alpha = 1, \beta = -1, r = 5$, and thus $\alpha - \beta + r = 7$.

The correct answer is B (2) because by substituting the points into the general equation of a circle, we find that k = 2 is the only valid solution.

Option C ((3,-1)) is correct because the tangent to the first circle at (1,-2) is also tangent to the second circle, and (3,-1) satisfies both the tangent line equation and the second circle's equation after shifting.

The correct answer is $\frac{{r^4}}{{2|a b|}}$ because the area of the triangle is calculated using the intercepts of the tangent line on the coordinate axes.

Option C ((0,2q)) is correct because the set of possible values of p is (0, 2q), derived from the condition that the given line is tangent to the circle.

Option D ($\phi$) is correct because there is no value of $\lambda$ that satisfies both the condition for P to be an interior point and the condition that the circle neither touches nor intersects the axes.

Option A is correct because the radius of the circle is the square root of the constant term in the equation, which is $\sqrt{36} = 6$.

The correct answer is D ($2x - 3y = - 13$) because it represents the equation of the chord bisected at the given point.

Option C ($x^2+y^2-2 x+2 y=47$) is correct because the center of the circle is found by solving the equations of the diameters, and the radius is calculated from the area. Substituting these values into the circle equation yields the correct result.

Option B (Are perpendicular) is correct because the line connecting the origin to the intersection points are perpendicular.

The correct answer is $11 x-2 y-46=0$ because it is derived using the tangent equation formula $xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c = 0$.

Option C ($\frac{{\sqrt 7 }}{3}$) is correct because after substituting $\lambda = 3$ into the given equation and simplifying, the radius of the resulting circle is calculated to be $\frac{{\sqrt 7 }}{3}$.

Option D ($P$ lies inside $C$ but outside $E$) is correct because substituting the coordinates of P into the equations of the ellipse and circle shows that P lies inside the circle and outside the ellipse.

Option A (4) is correct because the length of the chord is calculated to be 4 using trigonometric identities and the distance formula.

Option C (Intersect at two points) is correct because the distance between the centers of the circles is less than the sum of their radii, indicating they intersect.

The correct answer is A (${x^2} + {y^2} - 6x + 6y + 9 = 0$) because it represents a circle with radius 3 in the fourth quadrant touching both axes.

Option D ($1 / \sqrt{2}$ units) is correct because the radius is calculated using the given conditions and the equation of the circle.

Option D (All of these) is correct because the line always intersects the circle, meaning the discriminant of the quadratic equation formed by substituting the line equation into the circle equation is always non-negative for all values of m.

Option A ($4 y+3 x+22=0$) is correct because it represents the equation of the incident ray, considering the reflection from the line $x=-2$ and the tangency to the circle $x^2+y^2=4$.

Option C ((5,2)) is correct because the center of the circle must be equidistant from both lines, and solving the resulting equations gives the center's coordinates.