Question

If the radius of the circle x^2+y^2-2x+3y+k=0 is 5/2,then k=

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Value\:of\:k=-3}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies {x}^{2} + {y}^{2} -2x + 3y + k = 0 \\ \\ \tt: \implies Radius \: of \: circle = \frac{5}{2} \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies Value \: of \: k =?$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt: \implies Eqn \: of \: circle \to {x}^{2} + {y}^{2} + 2gx + 2fy + c= 0 - - - - - (1) \\ \\ \tt: \implies {x}^{2} + {y}^{2} - 2x + 3y + k = 0 \\ \\ \tt: \implies {x}^{2} + {y}^{2} + 2 \times( - 1 )\times x + 2 \times \frac{3}{2} \times y + k = 0 - - - - - (2) \\ \\ \text{Comparing \: (1) \: and \: (2)} \\ \tt \circ \: g = - 1 \\ \\ \tt \circ \: f = \frac{3}{2} \\ \\ \tt \circ \: c = k \\ \\ \bold{For \: Radius : } \\ \tt: \implies Radius = \sqrt{ {g}^{2} + {f}^{2} - c } \\ \\ \tt: \implies \frac{5}{2} = \sqrt{ {( - 1)}^{2} + (\frac{3}{2})^{2} - k } \\ \\ \tt: \implies (\frac{5}{2} )^{2} = 1 + \frac{9}{4} - k \\ \\ \tt: \implies k = 1 + \frac{9}{4} - \frac{25}{4} \\ \\ \tt: \implies k = \frac{4 + 9 - 25}{4} \\ \\ \tt: \implies k = \frac{ - 12}{4} \\ \\ \green{\tt: \implies k = - 3 }$

Answer 2

Question -

If the radius of the circle x^2+y^2-2x+3y+k=0 is 5/2,then k = ?

To Find -

The value Of k

Solution -

We know that -

The general equation of a circle can be represented by -

${x}^2 + {y}^2 + 2gx + 2fy + c = 0 \\ \\ \sf{ Comparing \: the \: coefficients \: : } \\ \\ => 2gx = -2x \\ \\ => g = -1 \\ \\ => 2fy = 3y \\ \\ => f = \dfrac{3}{2} \\ \\ => c = k$

Now we know the formula for the radius of a circle  -

$Radius \: Of \: A \: Circle =\sqrt{g^{2}+ f^{2} - c } \\ \\  \sf{ Substuting \: The \:  required \: values \: - } \\ \\ Radius = \sqrt{ 1 + \dfrac{9}{4} - k } = \dfrac{5}{2} \\ \\ \sf{ Squaring \: Both \: sides \: - } \\ \\ \dfrac{13}{4} - k = \dfrac{25}{4} \\ \\ - k = \dfrac{25}{4} - \dfrac{13}{4} = 3 \\ \\ k = -3 \ ........ [ A ]$

So the required value of k comes out to be -3.

Hence this is the required Answer .

Answer -

If the radius of the circle x^2+y^2-2x+3y+k=0 is 5/2,then k =  -3.