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- WE WANT TO FIND THE POINT IN THE FIRST QUADRANT
WHERE THE LINE Y = 2X INTERSECTS A CIRCLE WITH RADIUS 4
CENTERED AT THE ORIGIN.
LET'S BEGIN BY DETERMINING THE EQUATION OF THE CIRCLE.
WHERE IF THE EQUATION OF THE CIRCLE IS IN THIS FORM HERE
(H, K) WOULD BE THE CENTER AND R WOULD BE THE RADIUS.
SO BECAUSE OUR CIRCLE IS CENTERED AT THE ORIGIN
WHERE THE ORIGIN HAS COORDINATES (0, 0) NOTICE IN OUR CASE
BOTH H AND K ARE EQUAL TO 0 AND THE RADIUS IS 4 SO R = 4.
SO IF BOTH H AND K ARE ZERO
THIS WOULD JUST BE X SQUARED + Y SQUARED = 4 SQUARED OR 16.
THAT WOULD BE THE EQUATION OF THE CIRCLE
AND THE EQUATION OF THE LINE IS Y = 2X.
SO TO FIND THE POINT OF INTERSECTION
IN THE FIRST QUADRANT WE'LL SOLVE THIS
AS A SYSTEM OF EQUATIONS
BUT BEFORE WE DO THIS LETS LOOK
AT THE GRAPH OF THE CIRCLE AND THE LINE.
NOTICE HOW THE CIRCLE IS CENTERED AT THE ORIGIN
WITH A RADIUS OF 4 GIVING US
THE EQUATION X SQUARED + Y SQUARED = 16.
AND THE GIVEN LINEAR EQUATION WAS Y = 2X
SO NOTICE HOW THE Y INTERCEPT OF THIS LINE IS 0 HERE
AND BECAUSE THE SLOPE IS 2 OR 2/1
NOTICE IF WE GO UP 2 AND RIGHT 1 WE FIND ANOTHER POINT
ON THE LINE.
AND BECAUSE WE'RE LOOKING
FOR THE POINT OF INTERSECTION IN THE FIRST QUADRANT
WE'RE LOOKING FOR THIS POINT HERE.
SO GOING BACK TO OUR SYSTEM OF EQUATIONS
LETS SOLVE THIS USING SUBSTITUTION
WHERE IF Y = 2X WE'LL SUBSTITUTE 2X FOR Y IN THE FIRST EQUATION.
THIS WILL GIVE US AN EQUATION WITH JUST X.
WE WOULD HAVE X SQUARED + INSTEAD OF Y SQUARED
WE'D HAVE 2X SQUARED = 16.
2X TO THE 2nd POWER WOULD BE 4X SQUARED
SO WE HAVE X SQUARED + 4X SQUARED = 16.
X SQUARED + 4X SQUARED OR 1X SQUARED + 4X SQUARED
WOULD BE 5X SQUARED.
ISOLATE X SQUARED BY DIVIDING BOTH SIDES BY 5
SO WE HAVE X SQUARED = 16/5.
NOTICE HERE X CAN BE BOTH POSITIVE OR NEGATIVE
SO NOW WHEN WE TAKE THE SQUARE ROOT
OF BOTH SIDES OF THE EQUATION
WE NEED TO BE SURE TO INCLUDE A PLUS OR MINUS SIGN
ON THE RIGHT TO OBTAIN BOTH SOLUTIONS.
SO WE HAVE X = + OR - THE SQUARE ROOT OF 16
DIVIDED BY THE SQUARE ROOT OF 5.
BUT SINCE THE SQUARED OF 16 = 4 WE HAVE X = + OR - 4
DIVIDED BY SQUARE ROOT 5.
NOW BECAUSE WE'RE LOOKING FOR THE POINT
IN THE FIRST QUADRANT WE KNOW X HAS TO POSITIVE
SO FOR OUR PURPOSES WE KNOW THAT X CAN ONLY BE 4
DIVIDED BY SQUARE ROOT 5.
NOW YOU MAY BE ASKED TO RATIONALIZE THIS
BUT WE'LL SHOW THAT IN A FEW MOMENTS.
RIGHT NOW, WE JUST FOUND THAT THE POINT OF INTERSECTION
IN THE FIRST QUADRANT HAS AN X COORDINATE OF 4
DIVIDED BY SQUARE ROOT 5.
NOW LET'S FIND THE Y COORDINATE OF THE POINT OF INTERSECTION.
WELL WE KNOW Y MUST = 2 x X OR Y = 2X.
SO IF Y = 2X AND NOW WE KNOW THE VALUE OF X WE CAN FIND Y.
Y WOULD BE EQUAL TO 2 OR 2/1 x 4 DIVIDED BY SQUARE ROOT 5
WHICH WOULD BE 8 DIVIDED BY SQUARE ROOT 5.
SO THIS WOULD BE THE Y COORDINATE
OF THE POINT OF INTERSECTION IN THE FIRST QUADRANT,
NOW LETS SHOW HOW TO RATIONALIZE THESE COORDINATES
AND ALSO HOW TO GET A DECIMAL APPROXIMATION.
SO AGAIN, WE JUST FOUND THE X COORDINATE IS 4
DIVIDED BY SQUARE ROOT 5 AND THE Y COORDINATE IS 8
DIVIDED BY SQUARE ROOT 5.
SO IF WE WANT TO RATIONALIZE 4 DIVIDED BY SQUARE ROOT 5
WE'D HAVE TO MULTIPLY THE NUMERATOR AND DENOMINATOR
BY SQUARE ROOT 5 WHICH WOULD GIVE US
4 SQUARE ROOT 5 DIVIDED BY SQUARE ROOT 5 x SQUARE ROOT 5,
WHICH IS THE SQUARED OF 25 OR JUST 5.
SO AGAIN, THIS WAS X AND THEN Y,
REMEMBER Y = 2 x X SO 2 x 1 x 4 SQUARE ROOT 5/5
WOULD JUST BE 8 SQUARE ROOT 5/5.
SO IF WE ARE REQUIRED TO RATIONALIZE THE DENOMINATORS
OF THESE COORDINATES WE WOULD SAY
4 SQUARE ROOT 5 DIVIDED BY 5, 8 SQUARE ROOT 5 DIVIDED BY 5
AND ALSO JUST IN CASE YOU ARE ASKED TO ROUND THESE VALUES
TO LETS SAY 3 DECIMAL PLACES LETS ALSO SHOW THAT.
OF COURSE, THIS WOULD REQUIRE A CALCULATOR
AND IT WOULD BE EASIER TO USE THESE COORDINATES
TO CONVERT TO A DECIMAL.
SO TO CONVERT THE X COORDINATE TO A DECIMAL
WE WOULD HAVE 4 DIVIDED BY--
ON THIS CALCULATOR A 2nd X SQUARED
BRINGS UP THE SQUARE ROOT, 5 CLOSED PARENTHESIS ENTER.
ROUNDING TO 3 DECIMAL PLACES NOTICE
HOW THE 8 IN THE 4th DECIMAL PLACE
WOULD TELL US TO ROUND UP.
IT WOULD BE APPROXIMATELY 1.789.
AND NOW FOR THE Y COORDINATE
WE WOULD JUST HAVE 8 DIVIDED BY SQUARE ROOT 5.
NOTICE THERE'S A 7 IN THE 4th DECIMAL PLACE
SO AGAIN WE ROUND UP.
THIS WOULD BE APPROXIMATELY 3.578.
IF WE WANT TO CHECK THIS GRAPHICALLY,
THE DECIMAL APPROXIMATION WOULD PROBABLY BE THE BEST COORDINATES
TO USE.
NOTICE HOW THIS POINT OF INTERSECTION DOES LOOK
LIKE THE X COORDINATE IS A LITTLE BIT LESS THAN 2
AND THE Y COORDINATE IS A LITTLE BIT MORE THAN 3.5.
SO THE GRAPH DOES VERIFY OUR WORK.
I HOPE YOU FOUND THIS EXPLANATION HELPFUL.