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Slope Form of Tangent

What is the l...

Question

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

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Solution

The correct option is B

Let O be the centre of the circle and T be tha point at which tangent is drawn.

$△ P T O$ is right angled triangle.

Circles equation is,

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

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

$i . e ., (x + g)^{2} + (y + f)^{2} = g^{2} + f^{2} - c$

$∴$ Comparing with the standard form.

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

Radius $\equiv \sqrt{g^{2} + f^{2} - c}$

Since $△ P T O$ is right angled

$P T^{2} = P O^{2} - O T^{2}$

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

$= (x_{1} + g)^{2} + (y^{1} + f)^{2} - (g^{2} + f_{2} - c)$

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

$∴ P T = \sqrt{x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + c}$

Suggest Corrections

Similar questions

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

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

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

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

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