The length of the common chord of the circles $(x - a)^{2} + y^{2} = a^{2}$ and $x^{2} + (y - b)^{2} = b^{2}$

\frac{a b}{\sqrt{a^{2} + b^{2}}}

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

\frac{2 a b}{a^{2} + b^{2}}

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

\frac{a b}{a + b}

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

\frac{2 a b}{\sqrt{a^{2} + b^{2}}}

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

Open in App

Solution

The correct option is B $\frac{2 a b}{\sqrt{a^{2} + b^{2}}}$
[ From point (1) ]
$S_{1} : (x - a)^{2} + y^{2} = a^{2}$
$C_{1} \equiv (a, 0), r_{1} = a$
$S_{2} : x^{2} + (y - b)^{2} = b^{2}$
$C_{2} \equiv (0, b) r_{2} = b$
$∴ e q^{n}$ of chord is $S_{1} - S_{2} = 0$
line $A B : b y = a x - - - - (1)$
So, $C_{1} P = ∣ ∣ ∣ \frac{a^{2}}{\sqrt{a^{2} + b^{2}}} ∣ ∣ ∣$
In $Δ A C_{1} P,$
$A P = \sqrt{a^{2} - \frac{a^{4}}{a^{2} + b^{2}}}$
$= \frac{a b}{\sqrt{a^{2} + b^{2}}}$
So, length of chord is
$2 A P = \frac{2 a b}{\sqrt{a^{2} + b^{2}}}$

Suggest Corrections

Similar questions

The length of common chord of the circles $(x - a)^{2} + y^{2}$ = $a^{2}$ and $x^{2} + (y - b)^{2}$ = $b^{2}$ is

Q. The radius of the circle passing through the point of intersection of ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

and

x^{2} - y^{2} = 0

The length of common chord of the circles $(x - a)^{2} + y^{2}$ = $a^{2}$ and $x^{2} + (y - b)^{2}$ = $b^{2}$ is

Which of the following is correct?
a) $(a - b)^{2} = a^{2} + 2 a b - b^{2}$
b) $(a - b)^{2} = a^{2} - 2 a b + b^{2}$
c) $(a - b)^{2} = a^{2} - b^{2}$
d) $(a + b)^{2} = a^{2} + 2 a b - b^{2}$

Q. Question 3

Which of the following is correct?
a)

(a - b)^{2} = a^{2} + 2 a b - b^{2}

(a - b)^{2} = a^{2} - 2 a b + b^{2}

(a - b)^{2} = a^{2} - b^{2}

(a + b)^{2} = a^{2} + 2 a b - b^{2}

Join BYJU'S Learning Program

The length of the common chord of the circles (x−a)2+y2=a2 and x2+(y−b)2=b2

The correct option is B 2ab√a2+b2[ From point (1) ]S1:(x−a)2+y2=a2 C1≡(a,0),r1=aS2:x2+(y−b)2=b2 C2≡(0,b)r2=b∴eqn of chord is S1−S2=0lineAB:by=ax−−−−(1)So, C1P=∣∣∣a2√a2+b2∣∣∣In ΔAC1P,AP=√a2−a4a2+b2=ab√a2+b2So, length of chord is2AP=2ab√a2+b2

The length of the common chord of the circles $(x - a)^{2} + y^{2} = a^{2}$ and $x^{2} + (y - b)^{2} = b^{2}$