Circles practice · Easy

Option A ($4 x-6 y-13=0$) is correct because it represents the equation of the common chord obtained by subtracting the equations of the two circles.

Option D ($x^2+y^2-2 x-2 y+1=0$) is correct because the given line is tangent to the circle $(x-1)^2 + (y-1)^2 = 1$, which simplifies to the given equation.

Option C ((5,-2)) is correct because the equation of the circle is found using the given conditions, and substituting (5,-2) into the equation satisfies it.

Option D ($2 \sqrt{23}$) is correct because it is the length of the intercept made by the circle on the x-axis, calculated after finding the values of g and c.

Option A ($4 \leq x^2+y^2 \leq 64$) is correct because the locus represents the region between two circles centered at the origin with radii 2 and 8, respectively, due to the radius of the moving circle being 3 and the fixed circle having a radius of 5.

Option D ($x^2+13 x-4 y=1$) is correct. Reflecting a curve in the origin involves replacing x with -x and y with -y in the original equation.

Option A ($3x + 4y - 6 = 0$) is correct because the perpendicular distance from the center of the circle to this line equals the radius, indicating it's a tangent.

Option C (3) is correct because the distance between the centers of the circles equals the sum of their radii, indicating that the circles touch externally, hence having 3 common tangents.

Option B ((3,-1)) is correct because the tangent to the first circle at (1,-2) is also tangent to the second circle at (3,-1).

Option D (None of these) is correct because the range of $\lambda$ for which the equation represents a circle is $[-2, -1]$.

Option C ($x{x_1} + y{y_1} = x_1^2 + y_1^2$) is correct because it represents the equation of the chord of the circle with the given midpoint.

Option C ((2,2)) is correct because converting the polar equation to Cartesian coordinates and completing the square reveals the center of the circle.

Option C ($\frac{12}{5}$) is correct because it is the value of $r^2$ that satisfies the given condition: the chord length is equal to r.

Option B Touch each other internally is correct because the distance between the centers is equal to the difference of the radii of the two circles.

Option A ($ab > 0, c > 0$) is correct because for one circle to lie within the other, their intersection points must be imaginary and the origin must be outside both circles.

Option A ($h = \pm r$) is correct because the condition for perpendicular tangents from the origin leads to this relationship between h and r.

Option C ($- 1$) is correct because the slope of the tangent to the circle at point (h,h) is -1.

Option B (-3,-3) is correct because the point of contact is found by solving the equation of the normal line passing through the circle's center and the given line.

Option C $\left( {5,2} \right)$ is correct because the symmetrical point about a line is the image of the point in that line, and the tangent at the origin is $y=x$.

Option D ($24 x+7 y=0$) is correct because it represents the equation of the other equal side passing through the origin, found using the perpendicular distance from the center to the line.

Option A ($4\sqrt 2 - 5$) is correct because it represents the shortest distance between the parabola and the circle, calculated along the common normal.

Option C $((-3 \sqrt{2},-\sqrt{3}) \cup(\sqrt{3}, 3 \sqrt{2}))$ is correct because it represents the set of 'a' values that satisfy both conditions: the point lies outside the first circle and inside the second circle.

Option A ($(1 / 2,1 / 4)$) is correct because the chord of contact equation can be rearranged to represent a family of lines passing through the intersection of two fixed lines, which gives the required point.

The correct answer is $x+y=2 \sqrt{{2}}$ because the perpendicular distance from the center of the circle to the tangent must equal the radius.

Option D ($\pi / 2$) is correct because the origin lies on the director circle of the given circle.

Option C ($m \in(-2,-1 / 2)$) is correct because it represents the range of slopes for which the line does not intersect the circle, determined by ensuring the distance from the circle's center to the line is greater than the radius.

Option A ($2 g^{\prime}\left(g-g^{\prime}\right)+2 f^{\prime}\left(f-f^{\prime}\right)=c-c^{\prime}$) is correct because it represents the condition for the first circle to bisect the circumference of the second circle.

Option C (16) is correct because there are 16 integer values of 'a' that satisfy the condition that the radius of the circle does not exceed 5.

Option B ($x - y = 0$) is correct because it satisfies the condition that the intercepts made by the circle on L1 and L2 are equal.

Option B (7) is correct because the radius is found by converting the polar equation to Cartesian form and then calculating the radius of the resulting circle.