The circles x^2 + y^2 + 2gx + 2fy = 0 and x^2 | vedclass

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

Question

(A) For two circles with centers $C_1(-g, -f)$ and $C_2(-g', -f')$ and radii $r_1 = \sqrt{g^2 + f^2}$ and $r_2 = \sqrt{g'^2 + f'^2}$ to touch external

Vedclass · Accepted Answer

$f'g = g'f$

Vedclass · Answer

(A) For two circles with centers $C_1(-g, -f)$ and $C_2(-g', -f')$ and radii $r_1 = \sqrt{g^2 + f^2}$ and $r_2 = \sqrt{g'^2 + f'^2}$ to touch externally,the distance between their centers must be equal to the sum of their radii.$C_1C_2 = r_1 + r_2$$\sqrt{(-g + g')^2 + (-f + f')^2} = \sqrt{g^2 + f^2} + \sqrt{g'^2 + f'^2}$Squaring both sides:$(g' - g)^2 + (f' - f)^2 = (g^2 + f^2) + (g'^2 + f'^2) + 2\sqrt{g^2 + f^2}\sqrt{g'^2 + f'^2}$$g'^2 - 2gg' + g^2 + f'^2 - 2ff' + f^2 = g^2 + f^2 + g'^2 + f'^2 + 2\sqrt{(g^2 + f^2)(g'^2 + f'^2)}$$-2(gg' + ff') = 2\sqrt{(g^2 + f^2)(g'^2 + f'^2)}$$-(gg' + ff') = \sqrt{(g^2 + f^2)(g'^2 + f'^2)}$Squaring again:$(gg' + ff')^2 = (g^2 + f^2)(g'^2 + f'^2)$$g^2g'^2 + f^2f'^2 + 2gg'ff' = g^2g'^2 + g^2f'^2 + f^2g'^2 + f^2f'^2$$2gg'ff' = g^2f'^2 + f^2g'^2$$g^2f'^2 + f^2g'^2 - 2(gf')(fg') = 0$$(gf' - fg')^2 = 0$$gf' = fg'$

The circles $x^2 + y^2 + 2gx + 2fy = 0$ and $x^2 + y^2 + 2g'x + 2f'y = 0$ touch each other externally if:

Similar Questions

The centre of a circle which cuts $x^{2}+y^{2}+6x-1=0$,$x^{2}+y^{2}-3y+2=0$ and $x^{2}+y^{2}+x+y-3=0$ orthogonally is

The length of the diameter of the circle which cuts the following three circles orthogonally is:
$x^{2}+y^{2}-x-y-14=0$
$x^{2}+y^{2}+3x-5y-10=0$
$x^{2}+y^{2}-2x+3y-27=0$

The common tangent to the circles $x^2 + y^2 = 4$ and $x^2 + y^2 + 6x + 8y - 24 = 0$ also passes through the point

The equation of the circle described on the chord $3x + y + 5 = 0$ of the circle $x^2 + y^2 = 16$ as diameter is:

If $y = 2x$ is a chord of the circle $x^{2} + y^{2} = 10x$,then find the equation of the circle having this chord as its diameter.

Vedclass Test Series

Exam Paper Generator

Online Exam Module