If the two circles $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y = 0 & x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y = 0$ touch each then

f_{1} g_{1} = f_{2} g_{2}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{f_{1}}{g_{1}} = \frac{f_{2}}{g_{2}}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

f_{1} f_{2} = g_{1} g_{2}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

none

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B $\frac{f_{1}}{g_{1}} = \frac{f_{2}}{g_{2}}$
Since the 2 circles touch each other

$C_{1} C_{2} = r_{1} \pm r_{2}$

$⟹ \sqrt{(g_{1} - g_{2})^{2} + (f_{1} - f_{2})^{2}} = \sqrt{g_{1}^{2} + f_{1}^{2}} \pm \sqrt{g_{2}^{2} + f_{2}^{2}}$

$- 2 g_{1} g_{2} - 2 f_{1} f_{2} = \pm 2 (\sqrt{g_{1}^{2} + f_{1}^{2}}) (\sqrt{g_{2}^{2} + f_{2}^{2}})$

$(g_{1} g_{2})^{2} + (f_{1} f_{2})^{2} + 2 g_{1} g_{2} f_{1} f_{2} = (g_{1} g_{2})^{2} + (f_{1} f_{2})^{2} + (g_{1} f_{2})^{2} + (f_{1} g_{2})^{2}$

$2 = \frac{g_{1} f_{2}}{g_{2} f_{1}} + \frac{g_{2} f_{1}}{g_{1} f_{2}}$

Let $\frac{g_{1} f_{2}}{g_{2} f_{1}} = a$

$⟹ 2 = a + \frac{1}{a} ⟹ a^{2} + - 2 a + 1 = 0$

$⟹ a = 1 ⟹ \frac{g_{1} f_{2}}{g_{2} f_{1}} = 1$

$\frac{f_{1}}{g_{1}} = \frac{f_{2}}{g_{2}}$

Suggest Corrections

Similar questions

Q. If the two circles,

x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y = 0

and

x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y = 0

touches each other, then-

If the two circles $x^{2} + y^{3} + 2 g x + 2 f y = 0 a n d x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y = 0$ touch each other then

Q. If the circles

x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y = 0

and

x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y = 0

touch each other prove that:

f_{1} g_{2} = f_{2} g_{1}

Q. If two circles

x^{2} + y^{2} + 2 g x + 2 f y = 0

and

x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y = 0

touch each other, then

Two circles $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0 a n d x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c^{2} = 0$ are said to be orthogonal. Then $2 g_{1} g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2}$

Join BYJU'S Learning Program

If the two circles x2+y2+2g1x+2f1y=0&x2+y2+2g2x+2f2y=0 touch each then

If the two circles $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y = 0 & x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y = 0$ touch each then