Question

Topic: "Circles in the coordinate plane"
State the coordinates of the center and the measure of the radius; x^2+ y^2-16x=0.
What I know about my problem is that the formula for circles with center (h, k) and radius r is: (x-h)2+(y-k)2=r2

I know that I have to work backwards because the problem gives us the equation but i don't know how to solve it to get the center coordinates and the radius.

Answer 1

Answer:

TO GET THE COORDINATE OF CENTER AND RADIUS:–

${x}^{2} + {y}^{2} - 16x = 0$

ADD 64 IN BOTH SIDE –

${x}^{2} - 16x + 64 + {y}^{2} = 64$

${(x - 8)}^{2} + {y}^{2} = {8}^{2}$

NOW COMPARE WITH –

${(x - h)}^{2} + {(y - k)}^{2} = {r}^{2}$

SO ,

$h = 8 \:</strong><strong>,</strong><strong> \: k = 0 \: \: and \: \: r = 8$

☆☆☆ CENTER =》(8,0) AND RADIUS = 8 UNIT ☆☆☆

Answer 2

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

$\green{\tt{\therefore{Coordinate\:of\:centre=(8,0)}}}$

$\green{\tt{\therefore{Radius=8\:units}}}$

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

$\green{\underline \bold{Given : }} \\ \tt: \implies Eqn \: of \: circle = {x}^{2} + {y}^{2} - 16x = 0 \\ \\ \red{\underline \bold{To \: Find : }} \\ \tt: \implies Coordinate \: of \: centre = ? \\ \\ \tt: \implies Radius \: of \: circle = ?$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt: \implies {x}^{2} + {y}^{2} + 2gx + 2fy + c = 0 \\ \\ \tt \circ \: Eqn \: of \: standard \: circle \\ \\ \tt: \implies {x}^{2} + {y}^{2} - 16x = 0 \\ \\ \tt \circ \: given \: eqn \: of \: circle \\ \\ \text{So, \: it \: satisfy \: on \: stadard \: eqn} \\ \tt: \implies {x}^{2} + {y}^{2} - 16x + 64-64= 0 \\\\ \tt{:\implies (x-8)^2+y^{2}-64} \\\\ \tt{:\implies (x-8)^2+y^{2}=64}\\\\ \bold{Where:} \\ \tt \circ \: g = 0 \\ \\ \tt \circ \: f = 0 \\ \\ \tt \circ \: c =64 \\ \\ \bold{As \: we \: know \: that} \\ \\ \tt \circ Coordinate \: of \: centre \\ \tt: \implies x-8= 0 \\\\ \tt:\implies x=8 \\\\ \tt: \implies y = 0 \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies Radius = \sqrt{ {g}^{2} + {f}^{2} - c } \\ \\ \tt: \implies Radius = \sqrt{0 + 0 -(-64) } \\ \\ \tt: \implies Radius = \sqrt{64} \\ \\ \green{\tt: \implies Radius =8 \: units}$