Circles practice · Easy

Option A ($c=|a|$) is correct because the condition for touching circles (distance between centers equals sum or difference of radii) leads to this relationship.

Option D (1) is correct because the y-intercept is calculated using the formula 2*sqrt(f² - c) where f and c are coefficients from the circle's equation.

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.

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 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.

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.

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.

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 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 C ($\sqrt {{{a^2}(1 + {m^2})}} > c$) is correct because the condition for the line to intersect the circle in real points is that the discriminant of the quadratic equation formed by substituting the line equation into the circle equation must be greater than or equal to zero.

Option B (${x^2} + {y^2} - 30x - 10y + 225 = 0$) is correct because it represents a circle with radius 5, centered in the first quadrant, touching the x-axis and the line 4y = 3x.

Option C ((-4,0)) is correct because the range of g is determined by the condition that the point (α, α) lies outside the circle for all real α.

Option C ($\frac{10}{9} \sqrt{5}$ sq units) is correct because it is calculated using the length of the perpendicular from the center to the chord and the length of the chord.

Option B (-11) is correct because substituting the point (-3, 2) into the concentric circle equation allows us to solve for c.

Option A ((2,-3), 1) is correct because by completing the square, the center and radius of the circle $x^2+y^2-4 x+6 y+12=0$ are found to be (2, -3) and 1, respectively.