Circles practice · Medium

Option D ($\sqrt{22}$) is correct because by using properties of circles and similar triangles, we can calculate the radius of the circle.

Option D ({-1}) is correct because the line must be tangent to the circle, and only c=-1 satisfies the condition that the line contains only one point in common with the circle.

Option A (2x + 3y = 13, 2x - 3y = 13) is correct because substituting x=2 into the circle equation gives y=$\pm$3, and then using the tangent equation xx1 + yy1 = r² yields the correct tangent lines.

Option C ($9(x^2+y^2)=a^2$) is correct because the straight line touches the circle with radius a/3, derived from the condition that the pole lies on the circle $x^2+y^2=9a^2$.

Option C (19) is correct because the point C(a, a+1) must lie outside the circle x² + y² = 4, leading to 19 integral values for 'a' within the given range.

Option A ($2 x^2+2 y^2-a x-a y=0$) is correct because it represents the circle with the common chord of the given circles as its diameter.

Option D ($x^2+y^2-2 x-2 y+1=0$) is correct because it satisfies the given conditions for S(x,2)=0 and S(1,y)=0.

Option C ($p^2-q^2=0$) is correct because the tangents from the origin are perpendicular only when the circle touches both x and y axes, implying $p = \pm q$.

Option A ($(8, \infty)$) is correct because the condition for the line to bisect two distinct chords leads to the inequality $a^2 > 8$.

The correct answer is B (${x^2} + {y^2} - ax = 0$) because it represents the locus of the center of the circle drawn on the chord as diameter.

Option B ($6 + 4\sqrt 2$) is correct because it is the solution to the quadratic equation derived from the geometry of the two touching circles.

The correct answer is D. The line being normal to the circle implies it passes through the center. Only option D has a center that satisfies the normal's equation and the given point lying on the circle.

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 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.

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 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 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 ($\sqrt{6}$) 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.

Option C (-2) is correct because solving the equation $\mathrm{SS}_1=\mathrm{T}^2$ with $\mathrm{y}=0$ gives a quadratic equation whose sum of roots is -2.

The correct answer is B (15/2) because when two chords bisect each other, they are diameters, implying equal lengths, which allows solving for 'a' and then finding the radius.

Option A ($\left\{(x, y): x^2=4 y\right\} \cup\{(0, y): y<0\}$) is correct because it represents the locus of the center of the circle touching the given circle externally and the x-axis.

Option B (Both circles centres lie inside region of one another) is the incorrect statement because the centers of each circle lie on the circumference of the other, not inside.

Option B (1+4/sqrt(5), 2+8/sqrt(5)) is correct because it represents the point on the circle farthest from the origin, found by moving along the line connecting the origin and the circle's center.

Option B ($2 x^2-y^2=16$) is correct because the locus is found by using the equation of the tangent to the circle and the condition on the product of the slopes.

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 ($\frac{3 \sqrt{5}}{5}$) 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.