- WE WANT TO FIND THE STANDARD EQUATION OF THE ELLIPSE
THAT HAS A CENTER OF (5,4), A FOCUS OF (1,4),
AND A VERTEX OF (0,4).
SO TO FIND THE STANDARD EQUATION OF AN ELLIPSE
WE'LL BE USING ONE OF THESE TWO FORMS OF THE EQUATION
BASED UPON WHETHER THE ELLIPSE HAS A VERTICAL MAJOR AXIS
OR HORIZONTAL MAJOR AXIS.
IN BOTH CASES A IS GREATER THAN B,
AND THEREFORE A SQUARED IS A LARGER DENOMINATOR.
SO IF A SQUARED IS UNDER THE Y PART OF THE EQUATION,
WE HAVE A VERTICAL MAJOR AXIS.
AND IF A SQUARED IS UNDER THE X PART OF THE EQUATION,
WE HAVE A HORIZONTAL MAJOR AXIS.
SO TO FIND THE EQUATION, WE HAVE TO FIND H AND K,
WHICH ARE THE COORDINATES OF THE CENTER,
AND A SQUARED AND B SQUARED WHERE A IS THE DISTANCE
FROM THE CENTER TO THE TWO ENDPOINTS OF THE MAJOR AXIS
AND B IS THE DISTANCE FROM THE CENTER
TO THE TWO ENDPOINTS OF THE MINOR AXIS.
SO RIGHT AWAY, NOTICE THAT WE'RE GIVEN THE CENTER OF THE ELLIPSE,
WHICH MEANS H IS = TO 5 AND K IS = TO 4.
AND NOW TO FIND A SQUARED AND B SQUARED,
LET'S GO AHEAD AND PLOT THE GIVEN INFORMATION
ON THE COORDINATE PLANE AND ANALYZE THE RESULTS.
SO OUR CENTER HAS COORDINATES (5,4), WHICH WOULD BE HERE,
THE FOCUS HAS COORDINATES (1,4), WHICH WOULD BE HERE.
NOW NOTICE HOW THIS FOCUS IS ONE, TWO, THREE, FOUR UNITS
TO THE LEFT OF THE CENTER.
SO THIS IS THE CENTER AND THIS IS ONE FOCUS.
SO THE OTHER FOCUS MUST BE FOUR UNITS
TO THE RIGHT OF THE CENTER, WHICH WOULD BE HERE.
NEXT WE'RE TOLD THAT A VERTEX OF THE ELLIPSE
HAS COORDINATES (0, 4).
THE VERTICES OF THE ELLIPSE ARE THE ENDPOINTS OF THE MAJOR AXIS.
SO THE POINT (0, 4) WOULD BE HERE.
SO IF WE HAVE ONE VERTEX FIVE UNITS TO THE LEFT OF THE CENTER,
THE OTHER VERTEX MUST BE FIVE UNITS
TO THE RIGHT OF THE CENTER, WHICH WOULD BE HERE.
SO RIGHT AWAY, THIS TELLS US THAT OUR ELLIPSE
HAS A HORIZONTAL MAJOR AXIS,
THEREFORE WE'LL BE USING THIS FORM OF THE STANDARD EQUATION
OF AN ELLIPSE WHERE THE LARGER DENOMINATOR
WILL BE UNDER THE X PART OF THE EQUATION.
NEXT, THE DISTANCE FROM THE CENTER TO THE ENDPOINTS
OF THE MAJOR AXIS IS EQUAL TO A UNITS,
AND THEREFORE FOR THIS EXAMPLE
NOTICE HOW THE DISTANCE FROM THE VERTEX
TO THE CENTER IS FIVE UNITS, SO WE KNOW A = 5.
AND ALSO THE DISTANCE FROM THE CENTER TO THE TWO FOCI
IS EQUAL TO C UNITS.
SO NOTICE FOR OUR ELLIPSE C WOULD BE = TO 4.
BUT NOTICE TO WRITE THE EQUATION WE NEED B SQUARED,
AND THEREFORE WE HAVE TO FIND B.
AND WE CAN DO THIS USING THE FORMULA
A SQUARED = B SQUARED + C SQUARED,
WHICH IS ALWAYS TRUE FOR AN ELLIPSE.
SO IF WE WERE TO SOLVE THIS EQUATION FOR B SQUARED,
WE WOULD SUBTRACT C SQUARED ON BOTH SIDES.
SO WE CAN WRITE THE EQUATION B SQUARED = A SQUARED - C SQUARED.
AND SINCE WE HAVE A AND C, WE CAN NOW FIND B SQUARED.
B SQUARED IS EQUAL TO A SQUARED.
WELL, IF A IS 5, 5 SQUARED IS 25 - C SQUARED,
WHERE C IS 4.
4 SQUARED IS 16,
SO WE HAVE B SQUARED = 25 - 16 IS = TO 9.
SO NOW WE HAVE ALL THE INFORMATION WE NEED.
WE HAVE H AND K.
WE FOUND A = 5,
AND THEREFORE A SQUARED IS = TO 25.
AND WE JUST FOUND B SQUARED = 9.
AND, AGAIN, BECAUSE WE HAVE A HORIZONTAL MAJOR AXIS,
A SQUARED WILL BE UNDER THE X PART OF THE EQUATION.
SO OUR EQUATION'S GOING TO BE
THE QUANTITY X - H SQUARED OR X - 5 SQUARED
DIVIDED BY A SQUARED, WHICH IS 25,
+ THE QUANTITY Y - K SQUARED, WHICH WOULD BE Y - 4 SQUARED,
DIVIDED BY B SQUARED,
WHICH WE JUST FOUND TO BE 9 MUST = 1.
SO THIS WILL BE THE EQUATION OF OUR ELLIPSE
BASED UPON THE GIVEN INFORMATION.
BUT LET'S GO AHEAD AND FINISH BY GRAPHING THE ELLIPSE
ON THE COORDINATE PLANE.
TO FIND THE ENDPOINTS OF THE MINOR AXIS,
NOTICE HOW THE ENDPOINTS ARE B UNITS
ABOVE AND BELOW THE CENTER SINCE WE HAVE A HORIZONTAL MAJOR AXIS.
SO IF B SQUARED = 9
AND WE'RE ONLY CONCERNED ABOUT THE POSITIVE VALUE OF B,
WE SHOULD RECOGNIZE THAT B IS = TO +3,
WHICH MEANS THE TWO ENDPOINTS OF THE MINOR AXIS ARE THREE UNITS
ABOVE AND BELOW THE CENTER OF THE ELLIPSE.
SO ONE ENDPOINT OF THE MINOR AXIS WOULD BE HERE
AND THE OTHER ENDPOINT OF THE MINOR AXIS WOULD BE HERE
THREE UNITS BELOW THE CENTER.
WELL, THIS WOULD BE THE MINOR AXIS
AND THE ELLIPSE PASSES THROUGH THE ENDPOINTS OF THE MAJOR AXIS
AND MINOR AXIS,
AND THEREFORE WOULD LOOK LIKE THIS.
OKAY. I HOPE YOU FOUND THIS HELPFUL.