A circle C1 has radius 2 units and a circle C2 has radius 3 units. The distance between the centres of C1 and C2 is 7 units. If two lines, one tangent to both circles and the other passing through the center of both circles, intersect at point P which lies between the centers of C1 and C2 , then the distance between P and the centre of C2 is

Question

A circle C1 has radius 2 units and a circle C2 has radius 3 units. The distance between the centres of C1 and C2 is 7 units. If two lines, one tangent to both circles and the other passing through the center of both circles, intersect at point P which lies between the centers of C1 and C2 , then the distance between P and the centre of C2 is (A) (9/4) units (B) (7/3) units (C) (21/5) units (D) (14/5) units

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/m-wploy2t5qtdv.png" /> Since triangle O1PT1∼ triangle O2PT2 , ∴ (2/3)=(x/7 - x) 3x=14-2x x=(14/5) ⇒ PO2=7-(14/5)=(21/5) units

$\frac{9}{4}$ units

$\frac{7}{3}$ units

$\frac{21}{5}$ units

$\frac{14}{5}$ units

Since $△ O_{1} P T_{1} \sim △ O_{2} P T_{2}$ ,
$∴ \frac{2}{3} = \frac{x}{7 - x}$
$3 x = 14 - 2 x$
$x = \frac{14}{5}$
$\Rightarrow P O_{2} = 7 - \frac{14}{5} = \frac{21}{5}$ units

49​ units

37​ units

521​ units

514​ units

Since △O1​PT1​∼△O2​PT2​ , ∴32​=7−xx​ 3x=14−2x x=514​ ⇒PO2​=7−514​=521​ units

$\frac{9}{4}$ units

$\frac{7}{3}$ units

$\frac{21}{5}$ units

$\frac{14}{5}$ units

Since $△ O_{1} P T_{1} \sim △ O_{2} P T_{2}$ ,
$∴ \frac{2}{3} = \frac{x}{7 - x}$
$3 x = 14 - 2 x$
$x = \frac{14}{5}$
$\Rightarrow P O_{2} = 7 - \frac{14}{5} = \frac{21}{5}$ units