Circles practice · Mixed difficulty

Option C ($x-y=0$) is correct because the center of the circle lies on the line equidistant from the two parallel tangent lines.

Option B ($\frac{{192}}{{25}}\;\mathrm{sq}.\mathrm{units}$) is correct because it is calculated using the formula for the area of the triangle formed by the tangents from a point to a circle and the chord of contact.

Option B ($x^2+y^2-6 x-2 y=0$) is correct because it represents the circumcircle of the triangle, found by using the reflection of the orthocenter in the line BC.

Option A (2, 3) is correct because these side lengths satisfy the area condition and the cyclic quadrilateral properties.

Option B ($x^2+y^2-2 x+4 y-11=0$) is correct because it represents the equation of a circle with center (1, -2) and radius 4.

Option D (65) is correct because it is the sum of the product of all possible values of 'a' and the product of all possible values of 'b', where (a, b) lies on the parabola and the tangent at that point passes through the center of the given circle.

Option C ($x^2+y^2-2 x+2 y-47=0$) is correct because the center of the circle is found by solving the diameters' equations, and the radius is derived from the given area.

Option D ($\sqrt{c_1-c}$) is correct because it is the length of the tangent calculated using the formula $\sqrt{S_1}$.

Option A (12) is correct because by comparing the equation of the circle with the given information, we find a = -2, b = 1, and c = -13, leading to a+b-c = 12.

The correct answer is Collinear because the determinant formed by the coordinates of the centers of the three circles is zero, indicating that they lie on the same line.

The correct answer is C (2) because the length of the common tangent is the sum of the lengths of the tangents from the center of similitude to each circle.

Option B ((-7,8)) is correct because the locus of P is a circle with center (-7,8) after applying the given ratio condition.

Option A (2 < r < 8) is correct because it satisfies the condition for two circles to intersect at two distinct points, where the distance between the centers is between the difference and the sum of the radii.

Option A (2) is correct because it is the value of lambda that satisfies the condition for the circle to pass through the intersection points, derived from the given lines using the property a1*a2 = b1*b2.

Option B ($\frac{a-b}{h}=\frac{a^{\prime}-b^{\prime}}{h^{\prime}}$) is correct because it represents the condition for two curves to intersect in four concyclic points.

Option A (1) is correct because the product of the roots of the equation formed by the concyclic points is 1.

Option C ($x^2+y^2-2 x-6 y=0$) is correct because it represents a circle passing through the origin with its center on the line y=3x at a distance of $\sqrt{10}$ from the origin.

Option A ($x^2+y^2+x+2 y-5=0$) is correct because it represents a circle touching the given line at (1,1) and satisfies the condition S(2,3)=16.

Option C ($x^2+y^2-4 x+2 y+1=0$) is correct because the director circle of a circle has the same center but a radius that is $\sqrt{2}$ times the original circle's radius.

Option A ($2 < r < 8$) is correct because it represents the range of values for r that allow the two circles to intersect at two distinct points, based on the distance between their centers and their radii.

Option C ($x^2+y^2-2 x-4 y-20=0$) is correct because it represents the equation of a circle with center (1,2) and passing through (4,6), derived from the distance formula and the standard circle equation.

Option C (two values of $a$) is correct because substituting the tangency condition into the equation yields a quadratic equation in 'a', resulting in two possible solutions.

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

Option C (5) is correct because it represents the length of the tangent segment from the point Q on the y-axis to the circle.

Option A ($\sqrt{{\beta-\alpha}}$) is correct because it represents the length of the tangent calculated using the given conditions.

Option A ($x=y$) is correct because the normal to a circle passes through its center, and the line x=y passes through both the center and the given point.

The correct answer is a circle of radius 2 because the locus of P is the director circle of both circles.

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 radius can be found from the area.

Option A ($(x+16)^2+(y+2)^2=5^2$) is correct because it represents the correct transformation of the original circle across the given line mirror.

Option A ($\sqrt{6}$) is correct because it is the radius of circle C2 calculated using the distance from the center to the chord and half the chord length.