Circles practice · Easy

Option A (${x^2} + {y^2} - 4\left( {\sqrt 2 x + y} \right) + 4 = 0$) is correct because it satisfies the given conditions of touching the y-axis and having a specific x-intercept.

Option A ($x^2+y^2+3x-3y-a^2+9=0$) is correct because the equation of the circle on AB as diameter can be found by considering the family of circles passing through the intersection of the given circle and line, and then using the condition that the center of the circle lies on the given line.

Option D ($\pi / 2$) is correct because the product of the slopes of the two tangents from the origin to the circle is -1, indicating that the tangents are perpendicular.

Option A ${x^2} + {y^2} + 2x + 4y - 15 = 0$ is correct because it represents the circle with the given diameter endpoints derived from the roots of the provided equations.

Option A (1) is correct because the circles touch internally, resulting in only one common tangent.

Option B ($x^2+y^2-6 x-2 y+5=0$) is correct because it represents the equation of the circle where the given chord subtends an angle of $\pi/4$ at the circumference.

The correct answer is $3(x^2+y^2)=4 a^2$ because it is derived by eliminating the parameter between the tangent equations.

Option B ($52 r^2$) is correct because the condition for tangency (distance from center to line equals radius) is applied to find $k^2$.

Option A ((-7/4, 5/4)) is correct because it is the center of the circle formed by the intersection points of the given lines with the axes.

Option A ($4 x^2+4 y^2-20 x+4 y+8=0$) is correct because it represents the equation of a circle that touches the line y=x at (1,1) and has y=x-3 as a normal.

Option A ($x - 3y = 0$) is correct because it represents the other tangent from the origin to the circle, found by using the combined equation of tangents.

Option D ($4 x^2+4 y^2+8 a x+8 b y+3 a^2+3 b^2=0$) is correct because the equation of the incircle can be derived from the circumcircle's equation, considering the relationship between their centers and radii in an equilateral triangle.

Option B (an ellipse) is correct because the transformation scales one coordinate while leaving the other unchanged, resulting in an ellipse.

Option C ($x-2 y-5=0$) is correct because it represents the equation of the chord with the given midpoint, found using the property T = S1.

Option B (a parabola) is correct because the locus of the center is derived to be a parabola equation after eliminating the constant term.

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

Option B (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 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.