Circles practice · Easy

Option C ($\sqrt{109}$) is correct because it is the distance between the center of the circle and the point of intersection of the given lines, which equals the radius.

Option B ($x^2+y^2=x+y$) is correct because it represents the equation of the circle with AB as a diameter, where A and B are the intersection points of the given circle and the line y=x.

Option C ($a=\frac{{-4}}{{3}}, b=1$) is correct because by using the conditions of tangency and normality, we can solve for a and b.

Option C (2) is correct because the condition ${r_1} - {r_2} < {C_1}{C_2} < {r_1} + {r_2}$ is satisfied, indicating that the circles intersect at two points.

Option D ($m \in R$) is correct because the condition for the line to intersect the circle at two distinct points holds for all real values of m.

Option A (150) is correct because the area of the regular polygon is calculated using the given chord lengths and their distance apart to find the circle's radius, then applying the area formula for a regular polygon.

Option A ($\frac{{2b}}{{\sqrt {{a^2} - 4{b^2}} }}$) is correct because it is derived from the tangency condition of a line to a circle.

Option A ($- 35 < m < 15$) is correct because it satisfies the condition for the line to intersect the circle at two distinct points.

Option B (2) is correct because the distance between the centers of the two circles is less than the sum of their radii, indicating that they intersect, and thus have two common tangents.

Option D (None of these) is incorrect; the correct answer is C ($x^2+y^2+x-6=0$) which is found by using the given point and the condition that the center lies on the x-axis to solve for the circle's equation.

The correct answer is 1 because the radius is half the distance between the two parallel tangent lines.

Option C ($x^2+y^2-6 x-6 y+9=0$) is correct because it represents a circle touching both axes and passing through the point (3,6).

The correct answer is $4 x+3 y=25$ because the tangent equation is found by substituting the point into the general tangent equation and solving for the slope.

Option C (3) is correct because the length of the tangent is calculated by substituting the point's coordinates into the circle's equation and taking the square root.

Option C ($\sqrt{l_1^2+l_2^2}$) is correct because it represents the distance PQ, derived using the chord of contact and the lengths of tangents from P and Q.

Option C (${x^2} + {y^2} - 4x - 6y - 12 = 0$) is correct because it is the equation of the circle that satisfies the given conditions.

Option C ($x^2+y^2-2 x-4 y+4=0$) is correct because the circle's center lies at the intersection of the normals and the radius is the distance from the center to the tangent.

Option A ($5, - 35$) is correct because these are the values of k for which the line 3x - 4y + k = 0 is tangent to the given circle.

Option A (2) is correct because the shortest distance is the absolute difference between the distance from the point to the center and the radius of the circle.

Option C ($a = b$ and $h = 0$) is correct because for a general second-degree equation to represent a circle, the coefficients of $x^2$ and $y^2$ must be equal, and the coefficient of $xy$ must be zero.

The correct answer is C ($\frac{1}{{{a^2}}} + \frac{1}{{{b^2}}} = \frac{1}{{{c^2}}}$) because it represents the condition for the two circles to touch each other externally, derived from the distance between their centers and the sum of their radii.

Option A (A parabola) is correct because the locus of the center of a circle passing through a fixed point and touching the y-axis is a parabola.

Option C ($x + 2y = 3$) is correct because it represents the equation of the normal to the given circle at the point (1,1).

Option C ((3, -1)) is correct because it is the pole of the given line with respect to the given circle, calculated using the polar equation.

Option B ($4 \sqrt{3}$ units) is correct because the length of the tangent from a point to a circle is calculated using the formula $\sqrt{S_1}$, where S_1 is obtained by substituting the point's coordinates into the circle's equation.

Option B (These circles touch each other internally) is correct because the distance between their centers is equal to the difference of their radii.

Option C (2) is correct because the distance between the centers is less than the sum of the radii, indicating two common tangents can be drawn.

Option C ($x^2-y^2=0$) is correct because equal intercepts on the axes imply that the center of the circle lies on the lines $x = y$ or $x = -y$.

Option C ($x^2+y^2+gx+fy=0$) is correct because it represents the locus of the midpoint of the chords passing through the origin.

Option A ($c^2+6 c+7=0$) is correct because it represents the condition for the line to be tangent to the circle and perpendicular to the tangent of the other circle.