Circles practice · Mixed difficulty

Option C (7) is correct because after calculating the radius and using the formula for the length of the tangent, k is found to be 1525, so k-1520 = 5.

Option A (${x^2} + {y^2} - x - y = 0$) is correct because it represents the equation of the circle with AB as its diameter, where A and B are the intersection points of the line y=x and the given circle.

Option B (${x^2} + {y^2} - 18x - 16y + 120 = 0$) is correct because the point (5,5) lies on this circle and it has a radius of 5, touching the given circle at (5,5).

Option D (A straight line perpendicular to the line joining the centres of the circles) is correct because the locus of points from which the lengths of tangents to two circles are equal is the radical axis, which is a straight line perpendicular to the line joining the centers of the circles.

Option B (G.P.) is correct because the condition for the chord of contact to touch another circle leads to a geometric progression relationship between a, b, and c.

Option C ($\sqrt{2} a$) is correct because the locus of the point is a circle with radius $\sqrt{2}a$, derived by using the chord of contact and the condition that it subtends a right angle at the center.

Option B (be collinear) is correct because if the chords of contact from three points to a circle are concurrent, the points must be collinear, as shown by the determinant condition.

Option A ($2x - y \pm 5 = 0$) is correct because it represents the tangent line that satisfies the tangency conditions for both the circle and the parabola.

Option B ($p=3, q=2$) is correct because for the given equation to represent a circle, the coefficient of $x^2$ must equal the coefficient of $y^2$, and the coefficient of $xy$ must be zero.

Option C ($5\sqrt 2$) is correct because the distance between the intersection points A and B is calculated using the distance formula.

Option D (-3) is correct because by substituting the equation of the line into the circle's equation and setting the discriminant to zero (tangency condition), we can simplify the expression to -3.

Option C ($\left(4-\frac{5}{\sqrt{2}}, 4+\frac{5}{\sqrt{2}}\right)$) is correct because it represents one of the possible centers of the circle based on the given conditions.

Option B (${x^2} + {y^2} = x + y$) is correct because it represents the circle with the intercept AB as its diameter, found by substituting the intersection points into the general circle equation.

Option B (touch each other internally) is correct because the distance between the centers is equal to the difference of the radii, indicating internal tangency.

Option A (20) is correct because it is found by determining the value of k using the tangency condition, then finding the point of tangency (a, b) and summing k + a + b.

Option D ($2 \sqrt{5}$) is correct because the length of the intercept cut by the circle C on the x-axis is calculated to be $2 \sqrt{5}$ after finding the equation of the circle.

Option D ($256\pi$) is correct because it is the area between the two circles calculated using the given tangent length.

Option A ($3 x^{2}+3 y^{2}-2 \sqrt{3} a y=3 a^{2}$) is correct because it represents the circumcircle equation derived from the properties of an equilateral triangle and the general circle equation.

Option C (${x^2} + {y^2} - 2x + 2y - 47 = 0$) is correct because the center of the circle is the intersection of the diameters, and the equation is found using the center and radius.

Option B ($g^2+f^2=2 c$) is correct because the origin lies on the director circle of the given circle, which leads to this condition.

Option A ($4 x-6 y-13=0$) is correct because it represents the equation of the common chord obtained by subtracting the equations of the two circles.

Option D ($x^2+y^2-2 x-2 y+1=0$) is correct because the given line is tangent to the circle $(x-1)^2 + (y-1)^2 = 1$, which simplifies to the given equation.

Option C ((5,-2)) is correct because the equation of the circle is found using the given conditions, and substituting (5,-2) into the equation satisfies it.

Option D (\$a= \pm \sqrt{{2}} b$\$) is correct. By finding the slopes of the medians and setting their product to -1 (since they are perpendicular), the relationship between a and b is found to be$a= ± $\sqrt{{2}}$ b.

Option B (${x^2} + {y^2} - 7x - 10y + 22 = 0$) is correct because it represents a circle passing through P, Q, and (1, 2), derived using the chord equation and circle family concept.

Option D ($2 \sqrt{23}$) is correct because it is the length of the intercept made by the circle on the x-axis, calculated after finding the values of g and c.

Option A ($4 \leq x^2+y^2 \leq 64$) is correct because the locus represents the region between two circles centered at the origin with radii 2 and 8, respectively, due to the radius of the moving circle being 3 and the fixed circle having a radius of 5.

Option D ($x^2+13 x-4 y=1$) is correct. Reflecting a curve in the origin involves replacing x with -x and y with -y in the original equation.

Option A ($3x + 4y - 6 = 0$) is correct because the perpendicular distance from the center of the circle to this line equals the radius, indicating it's a tangent.

Option C (${x^2} + {y^2} - 3x - 3y = 0$) is correct because it represents the equation of the circle passing through the intersection points of the given circles and having its center at (3/2, 3/2).