Locus of mid point of chords of x 2- y 2-2 g x -2 f y - c -0 that pass through the origin- is A- x2-y2-2 g x-2 f y-c-0B- x2-y2-g x-f y-c-0C- x2-y2-g x-f y-0D- 2 x 2- y 2- g x - fy - c -0

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Equation of a Chord with a Given Middle Point

Locus of mid ...

Question

Locus of mid point of chords of $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ that pass through the origin, is

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

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

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

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2 (x^{2} + y^{2} + g x + f y) + c = 0

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Solution

The correct option is C $x^{2} + y^{2} + g x + f y = 0$
Let the mid point of chords of circle be $P (h, k) .$
Equation of this chord is $T = S_{1}$
$x h + y k + g (x + h) + f (y + k) + c = h^{2} + k^{2} + 2 g h + 2 f k + c .$
It passes through $(0, 0)$
$\Rightarrow h^{2} + k^{2} + g h + f k = 0$
Hence required locus is $x^{2} + y^{2} + g x + f y = 0$

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Similar questions

Equation of the pair of tangents drawn from the origin to the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ is

Which of the following equations represents a circle with centre $(g, f)$ and radius $\sqrt{g^{2} + f^{2} + c}$ ?

Q. Find the locus of mid point of chord of

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

that pass through the origin.

The equation of tangent to the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ at its point $(x_{1}, y_{1})$ is given by $x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c = 0$

A circle is of the form $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ and a pair of tangents are drawn from $(x_{1}, y_{1})$ to the circle.The combined equation of tangents is $S S_{1} = T^{2} .$

Where $S = x^{2} + y^{2} + 2 g x + 2 f y + c$

$S_{1} = x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + c$

$T = x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c$

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Equation of a Chord with a Given Middle Point

Standard XII Mathematics

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Locus of mid point of chords of x2+y2+2gx+2fy+c=0 that pass through the origin, is

The correct option is C x2+y2+gx+fy=0Let the mid point of chords of circle be P(h,k). Equation of this chord is T=S1 xh+yk+g(x+h)+f(y+k)+c=h2+k2+2gh+2fk+c. It passes through (0,0) ⇒h2+k2+gh+fk=0 Hence required locus is x2+y2+gx+fy=0

Locus of mid point of chords of $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ that pass through the origin, is