The angle between the tangents drawn from a p | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The given circle is $x^2 + y^2 + 4x - 2y - 4 = 0$. Comparing with $x^2 + y^2 + 2gx + 2fy + c = 0$,we get $g = 2, f = -1, c = -4$. The center is $C

Vedclass · Accepted Answer

$x^2 + y^2 + 4x - 2y - 31 = 0$

Vedclass · Answer

(A) The given circle is $x^2 + y^2 + 4x - 2y - 4 = 0$. Comparing with $x^2 + y^2 + 2gx + 2fy + c = 0$,we get $g = 2, f = -1, c = -4$. The center is $C(-g, -f) = (-2, 1)$ and the radius $r = \sqrt{g^2 + f^2 - c} = \sqrt{4 + 1 - (-4)} = \sqrt{9} = 3$. Let $P(h, k)$ be a point on the locus. The angle between the tangents is $60^{\circ}$,so the angle between the line joining the center to $P$ and the radius is $30^{\circ}$. In the right-angled triangle formed by the center,the point of contact,and $P$,we have $\sin(30^{\circ}) = \frac{r}{CP}$. $\frac{1}{2} = \frac{3}{CP} \implies CP = 6$. $CP^2 = 36$. $(h + 2)^2 + (k - 1)^2 = 36$. $h^2 + 4h + 4 + k^2 - 2k + 1 = 36$. $h^2 + k^2 + 4h - 2k - 31 = 0$. Replacing $(h, k)$ with $(x, y)$,the locus is $x^2 + y^2 + 4x - 2y - 31 = 0$.

The angle between the tangents drawn from a point $P$ to the circle $x^2 + y^2 + 4x - 2y - 4 = 0$ is $60^{\circ}$. The locus of $P$ is:

Similar Questions

The locus of the midpoint of the chord of contact of tangents drawn from a point on the line $4x - 5y = 20$ to the circle $x^2 + y^2 = 9$ is:

Three distinct points $A, B$,and $C$ are given in a two-dimensional coordinate plane such that for each point,the ratio of its distance from $(1, 0)$ to its distance from $(-1, 0)$ is equal to $\frac{1}{3}$. What is the circumcenter of triangle $ABC$?

The locus of a point which moves such that the sum of the squares of its distances from the three vertices of a triangle is constant,is a circle whose centre is at the

The locus of the middle points of chords of the circle $x^2 + y^2 - 2x - 6y - 10 = 0$ which pass through the origin is:

The locus of a point $(x, y)$ whose distance from the point $(-g, -f)$ is always $a$,where $k = g^2 + f^2 - a^2$,is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module