Applications

Translate Conic

Translate a conic by substituting shifted x and y expressions and read the new center.

Example

Shift a conic by substituting translated variables.

highlighted = computed this step

Step 1 — Start with the base circle

Start with the original circle.

x2+y2=25x^{2} + y^{2} = \hl{25}

Step 2 — Read the shift

Read the horizontal and vertical shift.

h=2k=-3h= \hl{2} \quad k= \hl{-3}

Step 3 — Substitute shifted variables

Substitute shifted x and y expressions.

x→x−2y→y+3x\to x- \hl{2} \quad y\to y+ \hl{3}

Step 4 — Translated equation

Write the shifted equation.

(x−2)2+(y+3)2=25(x- \hl{2} )^{2} + (y+ \hl{3} )^{2} = \hl{25}

Step 5 — Plot translated conic

Plot the translated conic.

Coordinate plot\text{Coordinate plot}
Translated conic plotCircle translated to center C(2, -3).C

Step 6 — New center

State the new center.

C=(2,-3)C=( \hl{2} , \hl{-3} )
translate-conic To translate by (h, k), substitute x -> x-h and y -> y-k in the original equation.