Circles practice · Mixed difficulty

Option A is correct because the condition for the line to intersect the circle at distinct or coincident points is that the perpendicular distance from the center of the circle to the line is less than or equal to the radius.

Option B (a circle) is correct because the locus of point P is derived to be a circle equation.

Option B ($x + 3y = 0$) is correct because the center of the given circle lies on this line, making it a diameter.

Option C ($\pm 8$) is correct because the equation of the chord of minimum length passing through (1,2) is found using T=S1, leading to x-y+8=0, and thus a+b+c = $\pm$8.

The correct answer is B ($15{x^2} + 15{y^2} - 94x + 18y + 55 = 0$) because this circle passes through the given points and touches the given line.

Option A ($3y = x$) is correct because it represents the other tangent to the circle passing through the origin, found by using the distance from the center to the tangent equals the radius.

Option B is correct because the area of the triangle ABC is derived using the midpoint of the chord and the radius of the circle.

Option D ($100 < C < 156$) is correct because the circle must neither intersect nor touch the coordinate axes, and the point of intersection of the lines must lie inside the circle, leading to the specified range for C.

Option A ($x^2+y^2-4 x-3 y=0$) is correct because it represents the circle with the line segment intercepted by the axes as its diameter.

The correct answer is 3 because it represents the length of the tangent from the origin to the circle, calculated after finding the term independent of x and determining the circle's equation.

Option B ($(18,42)$) is correct because it represents the range of lambda values for which exactly two common tangents can be drawn to the given circles, based on the distance between centers and the radii.

Option D (Touch each other externally) is correct because the distance between the centers of the circles is equal to the sum of their radii.

The correct answer is 3 because the radius of the larger circle is found using the distance formula and the radius of the smaller circle.

Option A ($\left| {xy} \right| < \frac{1}{8}$) is correct because it is derived from the condition that the point lies inside the circumcircle and applying the AM-GM inequality.

Option C ($2x + 2y = 5$) is correct because the locus of the center of the circle is found by expressing the center's coordinates in terms of a parameter and eliminating the parameter.

The correct answer is $1 / a$ because the locus of (h,k) is a circle with radius equal to the reciprocal of the original circle's radius.

Option B (3) is correct because after setting $\lambda$ to 0 to make the equation a circle, the radius is calculated as 3.

Option A ((-1,4)) is correct because the point (a, -a) lies inside the circle when a is in the interval (-1, 4).

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 B ($x^2+y^2-6 x-2 y+1=0$) is correct because the lines $x=3$ and $x-3y=0$ are normals to the circle, indicating the center is at (3,1) and the radius is 3.

Option B ((-2 / 5,2 / 5)) is correct because the locus of the point is a circle, and the center is found by applying the section formula and substituting into the original circle equation.

Option B ($5 x^2+5 y^2-8 x-14 y-32=0$) is correct because it represents the equation of a circle with radius 3 that touches the given circle at the specified point.

Option B (a pair of parallel lines) is correct because the locus represents lines parallel to the given line at a distance equal to the radius.

Option B ($\left( { - 7,\;8} \right)$) is correct because it is the center of the circle representing the locus of point P, derived from the given ratio of tangents.

Option B is correct because the equation of the circle passing through (0,0), (1,0), and (0,1) is $x^2 + y^2 - x - y = 0$. Substituting (2k, 3k) into this equation and solving for k yields k = 5/13.

Option A ($3/4$) is correct because the radius of the circle is half the distance between the two parallel tangent lines.

Option A ($5(x^2+y^2)-24x-18y+25=0$) is correct because it represents a circle that touches the given circle and passes through the given point.

Option B ($2\sqrt 2$) is correct because it is the distance between the midpoint of the line segment and the center of 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.