Circles practice · Medium

Option C ($\frac{1}{\sqrt{2}}$) is correct because it represents the distance between the intersection points of the circle and the line, calculated using the midpoint locus.

Option B ($x^2+y^2-4 x-2 y+4=0$) is correct because it represents the circle symmetrically opposite to the given circle with respect to the line x+y=2, found by reflecting the center.

Option B ($\left(-\frac{{24}}{{13}}, \frac{{29}}{{13}}\right)$) is correct because the radical center of the three circles is the orthocenter of the triangle, which is the intersection of two altitudes.

Option A ($x^2+y^2-10 x-6 y-66=0$) is correct because the radius of the circle is found to be 10, and the equation matches the given center (5,3).

Option C ($\lambda < - 1 - 2\sqrt 2$) is correct because it ensures that the point (sin θ, cos θ) always lies inside the given circle for any real number θ.

Option D ($x = 1$) is correct because it is the equation obtained by substituting the midpoint (1,0) into the chord equation formula.

Option B ($9{r^2} = {a^2}$) is correct because it relates the radius of the circle touched by the polar to the radius of the original circle and the circle on which the pole lies.

Option B ($a=c$) is correct because it represents the condition for the given line to be tangent to the given circle, derived by setting the discriminant of the resulting quadratic equation to zero.

Option A ($2x - y \pm 5 = 0$) is correct because it represents the tangent line that satisfies the tangency conditions for both the circle and the parabola.

Option C ($5\sqrt 2$) is correct because the distance between the intersection points A and B is calculated using the distance formula.

Option D (-3) is correct because by substituting the equation of the line into the circle's equation and setting the discriminant to zero (tangency condition), we can simplify the expression to -3.

Option B (${x^2} + {y^2} = x + y$) is correct because it represents the circle with the intercept AB as its diameter, found by substituting the intersection points into the general circle equation.

Option A ($3 x^{2}+3 y^{2}-2 \sqrt{3} a y=3 a^{2}$) is correct because it represents the circumcircle equation derived from the properties of an equilateral triangle and the general circle equation.

Option D (\$a= \pm \sqrt{{2}} b$\$) is correct. By finding the slopes of the medians and setting their product to -1 (since they are perpendicular), the relationship between a and b is found to be$a= ± $\sqrt{{2}}$ b.

Option B (${x^2} + {y^2} - 7x - 10y + 22 = 0$) is correct because it represents a circle passing through P, Q, and (1, 2), derived using the chord equation and circle family concept.

Option C (${x^2} + {y^2} - 3x - 3y = 0$) is correct because it represents the equation of the circle passing through the intersection points of the given circles and having its center at (3/2, 3/2).

Option A (cuts the given circle orthogonally) is correct because the condition for the chord of contact passing through Q leads to the orthogonality condition of the two circles.

Option A ((10,0)) is correct because it minimizes the area of the triangle formed by the tangents to the circle from a point on the x-axis and their y-intercepts.

Option B ($l_1^2+l_2^2$) is correct because when points are conjugate with respect to the circle, the square of the distance between them equals the sum of the squares of the lengths of the tangents from the points to the circle.

Option A ($-1 < m < \frac{3}{4}$) is correct because the origin divides AB internally when the line y=mx lies between the tangents at the origin to the two circles.

Option D (None of these) is incorrect, and Option A ($y = m(x - 1) - 2 \pm 3\sqrt{1 + m^2}$) is the correct equation of the tangent to the circle, which is not listed as an option. The provided answer is missing a minus sign and has an extra square.

Option A (all values of $p$) is correct because the equation of the circle passing through the intersection points P and Q exists for all values of p.

The correct answer is B ($x^2+y^2+g x+f y=0$) because the circumcircle of the triangle OPQ passes through the origin and the intersection of the given circle and the chord of contact.

Option D (4) is correct because it uses the orthogonality condition for two circles to find the relationship between their radii.

Option B ($L$ is a chord of $S$) is correct because the distance from the center of the circle to the line is less than the radius of the circle.

Option B (1) is correct because the circles touch each other internally, resulting in only one common tangent.

Option C ($(1 / 2,1 / 2)$) is correct because the common chord always passes through the intersection of two lines derived from the given circle equations.

Option A ($x^2-y^2=0$) is correct because the locus of the center is given by |x| = |y|, which simplifies to x² - y² = 0.

Option B (121) is correct because PA.PB is equal to the power of the point P with respect to the circle, which is calculated by substituting the coordinates of P into the circle's equation.

Option A ($x^2+y^2-6 x-4 y+12=0$) is correct because it represents the circle obtained by finding the mirror image of the center of the original circle with respect to the given line and keeping the same radius.