P is a variable point of the line L=0. Tangents are drawn to the circle x2+y2=4 from P to touch it at Q and R. The parallelogram PQSR is completed. If L ≡ 2 x+y-6=0, then the locus of circumcetre of triangle P Q R is -

Question

P is a variable point of the line L=0. Tangents are drawn to the circle x2+y2=4 from P to touch it at Q and R. The parallelogram PQSR is completed. If L ≡ 2 x+y-6=0, then the locus of circumcetre of triangle P Q R is - (A) 2 x-y=4 (B) 2 x+y=3 (C) x-2 y=4 (D) x+2 y=3

Tardigrade Admin · Accepted Answer

Parallelogram PQSR is a rhombus Let circumcentre of triangle PQR is (h, k) <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/a3c03fc9c3c5d8d114019d0c6a8a790b-.png" /> which is the middle point of CP ∴ P becomes (2 h , 2 k ) which satisfies the line 2 x+y-6=0 ∴ 2(2 h )+2 k -6=0 ∴ locus is 2 x+y-3=0

Q. $P$ is a variable point of the line $L = 0$ . Tangents are drawn to the circle $x^{2} + y^{2} = 4$ from $P$ to touch it at $Q$ and $R$ . The parallelogram PQSR is completed.
If $L \equiv 2 x + y - 6 = 0$ , then the locus of circumcetre of $△ PQR$ is -

$2 x - y = 4$

$2 x + y = 3$

$x - 2 y = 4$

$x + 2 y = 3$

Parallelogram PQSR is a rhombus
Let circumcentre of $△ PQR$ is $(h, k)$

which is the middle point of $CP$
$∴ P$ becomes $(2 h, 2 k)$ which satisfies the line $2 x + y - 6 = 0$
$∴ 2 (2 h) + 2 k - 6 = 0$
$∴$ locus is $2 x + y - 3 = 0$

Q. P is a variable point of the line L=0. Tangents are drawn to the circle x2+y2=4 from P to touch it at Q and R. The parallelogram PQSR is completed. If L≡2x+y−6=0, then the locus of circumcetre of △PQR is -

2x−y=4

2x+y=3

x−2y=4

x+2y=3