Circles practice · Mixed difficulty

Option A (100) is correct because the perpendicular distance from the center of the circle to the tangent line equals the radius, which leads to $(\lambda + \beta)^2 = 100$.

Option A ($x^2+y^2+14 x-12 y+76=0$) is correct. The image of the circle is found by finding the image of the center with respect to the line and keeping the same radius.

Option D ((-6, -7)) is correct because the tangent to the curve at (1,7) touches the circle at (-6,-7) after solving the equations of the tangent and the circle.

The correct answer is A (1) because it represents the length of the chord of the circle along the y-axis, calculated using the y-intercept formula.

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 ($\frac{3 \sqrt{5}}{5}$) is correct because it is the distance between point A and the center of the locus of point P, calculated using the section formula and the equation of the circle.

Option B (${c^2} - 34c + 64 = 0$) is correct because the area of the quadrilateral formed by the tangents and radii is used to derive a quadratic equation in terms of $c$, and this option is the correct quadratic.

Option C ($\frac{1}{\sqrt{2}}$) is correct because it represents the distance between the intersection points of the circle and the line, calculated using the midpoint locus.

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 ($x^2+y^2-4 x-2 y+4=0$) is correct because it represents the circle symmetrically opposite to the given circle with respect to the line x+y=2, found by reflecting the center.

Option B ($\left(-\frac{{24}}{{13}}, \frac{{29}}{{13}}\right)$) is correct because the radical center of the three circles is the orthocenter of the triangle, which is the intersection of two altitudes.

Option A ($x^2+y^2-10 x-6 y-66=0$) is correct because the radius of the circle is found to be 10, and the equation matches the given center (5,3).

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.

Option C ($\lambda < - 1 - 2\sqrt 2$) is correct because it ensures that the point (sin θ, cos θ) always lies inside the given circle for any real number θ.

Option D ($x = 1$) is correct because it is the equation obtained by substituting the midpoint (1,0) into the chord equation formula.

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 ($9{r^2} = {a^2}$) is correct because it relates the radius of the circle touched by the polar to the radius of the original circle and the circle on which the pole lies.

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 B ($a=c$) is correct because it represents the condition for the given line to be tangent to the given circle, derived by setting the discriminant of the resulting quadratic equation to zero.

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.

The correct answer is $\left( {1 + {m^2}} \right)\left( {{x^2} + {y^2}} \right) - 2a\left( {x + my} \right) = 0$ because it represents the equation of the circle with the given conditions.

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