The point $(x_{1}, y_{1})$ lies inside the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ , if $x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f g_{1} + c$ is less than zero

True

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False

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Solution

The correct option is A
True

From the figure we can see that, $P C < r$ . this is the condition for P to lie inside the circle.

The coordinates of P are $(x_{1}, y_{1})$ and

$C \equiv (- g, - f)$

$r = \sqrt{g^{2} + f^{2} - c}$

$P C < r$

$\Rightarrow \sqrt{(x_{1} + g)^{2} + (y_{1} + f)^{2}} < \sqrt{g^{2} + f^{2} - c}$

$\Rightarrow x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + c < 0$

$\Rightarrow$ The given statement is correct

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Similar questions

The point $(x_{1}, y_{1})$ lies inside the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ , if $x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f g_{1} + c$ is less than zero

What is the length of tangent from a point $P (x_{1}, y_{1})$ to the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0.$

Q. Assertion :If the point

(2, 4)

is interior to the circle

x^{2} + y^{2} - 6 x - 10 y + k = 0

and the circle does not cut the axes at any point, then

25 < k < 32

Reason: If the point

(x_{1}, y_{1})

lies inside the circle

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

then

{x_{1}}^{2} + {y_{1}}^{2} + 2 g x_{1} + 2 f y_{1} + c < 0

Q. If

θ

is the angle subtended by the circle

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

at the point

P (x_{1}, y_{1})

and

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

then

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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The point (x1,y1) lies inside the circle x2+y2+2gx+2fy+c=0, if x21+y21+2gx1+2fg1+c is less than zero

The correct option is A True From the figure we can see that, PC<r. this is the condition for P to lie inside the circle. The coordinates of P are (x1,y1) and C≡(−g,−f) r=√g2+f2−c PC<r ⇒√(x1+g)2+(y1+f)2<√g2+f2−c ⇒x21+y21+2gx1+2fy1+c<0 ⇒ The given statement is correct

The point $(x_{1}, y_{1})$ lies inside the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ , if $x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f g_{1} + c$ is less than zero