Circles practice · Easy

The correct answer is A parabola because the locus of the point P results in an equation of the form ${y^2} = kx$, which represents a parabola.

Option A ($y = 3x - 5$) is correct because it represents the equation of the line passing through the center of the circle and the given point.

Option A ($h=\pm r$) is correct because the condition for perpendicular tangents from the origin leads to the relationship $h^2 = r^2$.

Option A (${x^2} + {y^2} - 2x + 2y - 23 = 0$) is correct because it represents the equation of a circle with center (1, -1) and radius 5, derived from the intersection of diameters and the given circumference.

Option D ($\left(\frac{{4}}{{3}}, \frac{{5}}{{3}}\right) \frac{{4}}{{3}} \sqrt{{2}}$) is correct because it accurately represents the center and radius calculated from the given circle equation after converting it to standard form.

Option B ($(8,-9)$) is correct because it represents the vertex of the square farthest from the origin, calculated using the circle's center and radius.

Option C $\left(-\frac{{1}}{{10}}, \frac{{1}}{{10}}\right)$ is correct because the power of the point $(a, b)$ with respect to the circle is constant only when $a + b = 0$ and $6a - 4b + 1 = 0$.

Option D ($\sqrt {{a^2} + {b^2} + {p^2} + {q^2}}$) is correct because it's derived from the distance formula using the roots of the given quadratic equations.

Option C $\left(a_1 x+b_1 y+c_1\right)\left(a_2 x+b_2 y+c_2\right)-\left(a_1 b_2+a_2 b_1\right) x y=0$ is correct because it represents the equation of the circle passing through the points of intersection of the given lines with the coordinate axes, given the condition $a_1 a_2 = b_1 b_2$.

Option A (Statement I is True, statement II is True; statement II is a correct explanation for statement I) is correct because the locus of points from which perpendicular tangents can be drawn to the circle is the director circle, and (17,7) lies on it.

Option C ($90^{\circ}$) is correct because the director circle is the locus of points from which tangents to the circle are perpendicular.

Option C ($\pm 10$) is correct because it represents the values of $k$ for which the line $3x + y + k = 0$ is tangent to the circle ${x^2} + {y^2} = 10$.

Option B ($25 / 4$) is correct because it is the value of $\alpha^2$ that satisfies the given conditions for the circle.

Option C ($a p / (1-p^2)$) is correct because it represents the radius of the circle that is the locus of P.

Option B (50) is correct because it is the value of 4a + 2b when (1, a) and (b, 2) are conjugate points with respect to the given circle.

Option D ($\sqrt {20}$) is correct because the radius is the distance between the center of the circle and the point P(6,2), where the center is the intersection of the two diameters.

Option C (${x^2} + {y^2} - ax - ay = 0$) is correct because the circle passes through the vertices of the square when AB and AD are the axes of reference.

Option A ($0, -40/9$) is correct because these are the values of $\lambda$ for which the circle passes through the given points and touches the y-axis.

Option B (50 sq units) is correct because the area of the square inscribed in the circle is half the square of the circle's diameter.

Option A ($x^2+y^2-10 x-6 y-66=0$) is correct because the radius of the circle is calculated to be 10, and the equation is derived accordingly.

Option D is correct because the locus of the centers of the circles that touch the given circle externally and the x-axis is a parabola.

Option C, (4,1), is correct because it represents the midpoint of the chord, which lies on the line perpendicular to the chord passing through the center of the circle.

Option D (4) is correct because the distance between the centers of the circles is greater than the sum of their radii, indicating four common tangents.

The correct answer is C (-56) because the condition for a circle to bisect the circumference of another circle is that the radical axis passes through the center of the first circle.

Option C ($x^2+y^2-a x \pm a y+\frac{a^2}{4}=0$) is correct because it represents a circle touching both axes and the line x=a.

Option A (${\left( {x - p} \right)^2} = 4qy$) is correct because it represents the locus of the other end of the diameter through A.

Option D ($\left( {5/2,5/2} \right)$) is correct because the center of the circle passing through the concyclic points is found by equating the equation of the circle to zero and finding the center.

Option A ($1:2$) is correct because the ratio of the lengths of the tangents from a point on the circle to the other two circles is calculated to be $1:2$.

Option D (8) is correct because the length of the chord AB, where the circle intersects the x-axis, is calculated using the formula $2\sqrt{{g^2 - c}}$ after finding the equation of the circle.

Option C (1,4) is correct because the midpoint of the chord is found by solving the equation of the line perpendicular to the chord and passing through the circle's center.