Question

helppp i’ll give brainlest

Answer 1

Answer:

$\boxed{\textsf{ The center of the circle is \bf \bigg( -\dfrac{17}{2},0\bigg) .}}$

Step-by-step explanation:

A equation of circle is given to us . From the equation we need to find out the centre of the circle .The equation is :-

$\sf\implies x^2+y^2+17x+42=0$

Now we know the standard equation of a circle as , ( x - h )² + (y - k)² = r² , where

• (h , k) is the centre of the circle.
• r is the radius of the circle .

Now , let's simplify out the equation and convert it into standard form .

$\sf\implies x^2+y^2+17x+42=0$

Step 1: Subtract 42 from both sides :-

$\sf\implies x^2+y^2+17x+42-42=0-42\\\\\sf\implies x^2+y^2+17x = -42$

Step 2:Group out the like terms:-

$\sf\implies (x^2+17x)+y^2 = -42$

Step 3: Complete the square for x² :-

$\sf\implies \bigg( x^2+\dfrac{2\times 17x}{2} + \bigg(\dfrac{17}{2}\bigg)^2\bigg)+y^2 = -42+\bigg(\dfrac{17}{2}\bigg)^2\\\\\sf\implies \bigg( x + \dfrac{17}{2}\bigg)^2 + y^2 =\dfrac{289}{4}-42$

Step 4: Convert the RHS into whole square.

$\sf\implies \bigg( x + \dfrac{17}{2}\bigg)^2 + y^2 =\dfrac{289-168}{4} \\\\\sf\implies\bigg( x + \dfrac{17}{2}\bigg)^2 + y^2 =\dfrac{121}{4}\\\\\sf\implies \bigg( x + \dfrac{17}{2}\bigg)^2 + y^2 =\bigg(\dfrac{11}{2}\bigg)^2$

Step 6: Convert it into the standard form .

$\sf\implies \bigg\{ x - \bigg( - \dfrac{17}{2}\bigg) \bigg\}^2+ ( y - 0 )^2 = \bigg(\dfrac{11}{2}\bigg)^2$

Hence here we got our equation in the standard form . On comparing it to the standard form we get that ,

$\sf\implies \pink{ Center = \bigg( -\dfrac{17}{2},0\bigg) }$