Let PQ and MN be two straight lines touching | vedclass

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

Question

(D) The given circle is $x^{2}+y^{2}-4x-6y-3=0$. Comparing this with the general equation $x^{2}+y^{2}+2gx+2fy+c=0$,we get the centre $O(2, 3)$ and ra

Vedclass · Accepted Answer

$3(x^{2}+y^{2})-12x-18y-25=0$

Vedclass · Answer

(D) The given circle is $x^{2}+y^{2}-4x-6y-3=0$. Comparing this with the general equation $x^{2}+y^{2}+2gx+2fy+c=0$,we get the centre $O(2, 3)$ and radius $r = \sqrt{g^{2}+f^{2}-c} = \sqrt{2^{2}+3^{2}-(-3)} = \sqrt{4+9+3} = \sqrt{16} = 4$.Let $R(h, k)$ be the point of intersection of the tangents $PQ$ and $MN$. The line $OR$ bisects $\angle AOB$. Thus,$\angle AOR = \angle BOR = \frac{1}{2} \angle AOB = \frac{1}{2} (\pi/3) = \pi/6$ (or $30^{\circ}$).In the right-angled triangle $	riangle OAR$,we have $\cos(\angle AOR) = \frac{OA}{OR}$.$\cos(30^{\circ}) = \frac{r}{OR} \implies \frac{\sqrt{3}}{2} = \frac{4}{OR} \implies OR = \frac{8}{\sqrt{3}}$.Squaring both sides,$OR^{2} = \frac{64}{3}$.Since $O$ is $(2, 3)$ and $R$ is $(h, k)$,$OR^{2} = (h-2)^{2} + (k-3)^{2}$.So,$(h-2)^{2} + (k-3)^{2} = \frac{64}{3}$.$3(h^{2}-4h+4 + k^{2}-6k+9) = 64$.$3(h^{2}+k^{2}-4h-6k+13) = 64$.$3(h^{2}+k^{2}) - 12h - 18k + 39 = 64$.$3(h^{2}+k^{2}) - 12h - 18k - 25 = 0$.Replacing $(h, k)$ with $(x, y)$,the locus is $3(x^{2}+y^{2}) - 12x - 18y - 25 = 0$.

Let $PQ$ and $MN$ be two straight lines touching the circle $x^{2}+y^{2}-4x-6y-3=0$ at the points $A$ and $B$ respectively. Let $O$ be the centre of the circle and $\angle AOB=\pi/3$. Then the locus of the point of intersection of the lines $PQ$ and $MN$ is:

Similar Questions

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:

The number of circles touching the line $y - x = 0$ and the $y$-axis is

If the sum of the distances of a point from the origin and the line $x = 2$ is $4$,then its locus is

The locus of the centre of the circle which cuts off intercepts of length $2a$ and $2b$ from the $x$-axis and $y$-axis respectively,is

$A$ circle is drawn to cut a chord of length $2a$ units along the $X$-axis and to touch the $Y$-axis. The locus of the centre of the circle is

Vedclass Test Series

Exam Paper Generator

Online Exam Module