The length of the common chord of the circles | vedclass

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

Question

(C) The equation of the common chord is given by subtracting the two circle equations: $[(x - a)^{2} + y^{2} - c^{2}] - [x^{2} + (y - b)^{2} - c^{2}]

Vedclass · Accepted Answer

$\sqrt{4c^{2} - a^{2} - b^{2}}$

Vedclass · Answer

(C) The equation of the common chord is given by subtracting the two circle equations: $[(x - a)^{2} + y^{2} - c^{2}] - [x^{2} + (y - b)^{2} - c^{2}] = 0$ $x^{2} - 2ax + a^{2} + y^{2} - c^{2} - x^{2} - y^{2} + 2by - b^{2} + c^{2} = 0$ $-2ax + 2by + a^{2} - b^{2} = 0$ $2ax - 2by - a^{2} + b^{2} = 0 \dots (1)$ The distance $p$ from the center of the first circle $(a, 0)$ to the line $(1)$ is: $p = \frac{|2a(a) - 2b(0) - a^{2} + b^{2}|}{\sqrt{(2a)^{2} + (-2b)^{2}}} = \frac{|2a^{2} - a^{2} + b^{2}|}{\sqrt{4a^{2} + 4b^{2}}} = \frac{a^{2} + b^{2}}{2\sqrt{a^{2} + b^{2}}} = \frac{1}{2}\sqrt{a^{2} + b^{2}}$ The length of the common chord is $2\sqrt{c^{2} - p^{2}}$: $L = 2\sqrt{c^{2} - \left(\frac{\sqrt{a^{2} + b^{2}}}{2}\right)^{2}}$ $L = 2\sqrt{c^{2} - \frac{a^{2} + b^{2}}{4}}$ $L = \sqrt{4c^{2} - a^{2} - b^{2}}$

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

Similar Questions

The length of the common chord of the circles $x^2 + y^2 = 12$ and $x^2 + y^2 - 4x + 3y - 2 = 0$ is (in $\sqrt{2}$)

If the two circles $x^2+y^2-2x-6y+10-r^2=0$ and $x^2+y^2-8x+2y+8=0$ have a common chord of non-zero length,then

Tangents are drawn from the point $(-1, -4)$ to the circle $x^2 + y^2 - 2x + 4y + 1 = 0$. The length of the corresponding chord of contact is:

If $A, B$ are the points of contact of the tangents drawn from the point $P(-2, -3)$ to the circle $x^2+y^2-8x-10y+5=0$ and the chord $AB$ subtends an angle $\theta$ at $P$,then $\tan \theta =$

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

Vedclass Test Series

Exam Paper Generator

Online Exam Module