the abscissa of a point in the 4th quadrant is numerically three times its ordinate and is 10 units from (-2, 4). find the point. please explain how you arrive at your answer. thanks a lot!
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First, the terms abscissa and ordinate should be clarified. Abscissa refers to the x coordinate and Ordinate refers to the y coordinate. (As a hint to remember this, both the Abscissa and "x" are alphabetically first.)
So, the first sentence states: (3y, y) since 3 times the ordinate is 3 times the y coordinate. When you see the phrase "numerically three times its ordinate," that means you are looking only at the numbers, not the signs.
In the fourth quadrant, the x coordinate is positive and the y coordinate is negative, so you know it will have the form (+ , -).
10 units from your point means you can use the distance formula with your point.
so 10 = square root( (-2 - 3y)^2 + (4 - y)^2 )
To solve this, square both sides and you get
100 = (-2 - 3y)^2 + (4 - y)^2
Using FOIL, we get
100 = (4 + 12y + 9y^2) + (16 - 8y + y^2)
Removing parentheses and combining like terms gives us
100 = 10y^2 + 4y + 20
bringing 100 to the right hand side by subtracting 100 on both sides gives us
0 = 10y^2 + 4y - 80
Factoring out a 2 reduces this to
0 = 5y^2 + 2y - 40
we can factor this into
0 = (5y - 8) (y + 2)
setting each term in parentheses = 0, we have
5y - 8 = 0
5y = 8
y = 5/8
y + 2 = 0
y = -2
So, y is either 5/8 or -2
In the fourth quadrant, the y is negative, so we need y = -2.
The abscissa is three times this number (not the sign) so it is 6.
Therefore, the point 10 units from (-2,4) meeting the requirements from the first sentence is (6, -2)
So, the first sentence states: (3y, y) since 3 times the ordinate is 3 times the y coordinate. When you see the phrase "numerically three times its ordinate," that means you are looking only at the numbers, not the signs.
In the fourth quadrant, the x coordinate is positive and the y coordinate is negative, so you know it will have the form (+ , -).
10 units from your point means you can use the distance formula with your point.
so 10 = square root( (-2 - 3y)^2 + (4 - y)^2 )
To solve this, square both sides and you get
100 = (-2 - 3y)^2 + (4 - y)^2
Using FOIL, we get
100 = (4 + 12y + 9y^2) + (16 - 8y + y^2)
Removing parentheses and combining like terms gives us
100 = 10y^2 + 4y + 20
bringing 100 to the right hand side by subtracting 100 on both sides gives us
0 = 10y^2 + 4y - 80
Factoring out a 2 reduces this to
0 = 5y^2 + 2y - 40
we can factor this into
0 = (5y - 8) (y + 2)
setting each term in parentheses = 0, we have
5y - 8 = 0
5y = 8
y = 5/8
y + 2 = 0
y = -2
So, y is either 5/8 or -2
In the fourth quadrant, the y is negative, so we need y = -2.
The abscissa is three times this number (not the sign) so it is 6.
Therefore, the point 10 units from (-2,4) meeting the requirements from the first sentence is (6, -2)
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Abscissa = a
Ordinate = - 3a (Negative because it is in Q4)
Point (- 2, 4):
x = - 2
y = 4
Distance, d = 10
(x - a)² + [y - (- 3a)]² = d²
(x - a)² + (y + 3a)² = d²
(- 2 - a)² + (4 + 3a)² = (10)²
(a² + 4a + 4) + (9a² + 24a + 16) = 100
a² + 9a² + 4a + 24a + 4 + 16 = 100
10a² + 28a + 20 = 100
10a² + 28a = 100 - 20
10a² + 28a = 80
5a² + 14a = 40
5(a² + 14/5 a) = 40
a² + 2.8a = 40 / 5
a² + 2.8a = 8
a² + 2.8a = 8
a² + 2.8a + 1.96 = 8 + 1.96
(a + 1.4)² = 9.96
a + 1.4 = √9.96
a + 1.4 = ± 3.156
a = - 1.4 ± 3.156
If a = - 1.4 + 3.156,
a = 1.756
If a = - 1.4 - 3.156,
a = - 4.56
Feasible Domain: a > 0 | a in Q4
Since a is positive in Q4,
a = 1.756
and
- 3a = - 3(1.756)
- 3a = - 5.268
The point is (1.756, - 5.268)
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Ordinate = - 3a (Negative because it is in Q4)
Point (- 2, 4):
x = - 2
y = 4
Distance, d = 10
(x - a)² + [y - (- 3a)]² = d²
(x - a)² + (y + 3a)² = d²
(- 2 - a)² + (4 + 3a)² = (10)²
(a² + 4a + 4) + (9a² + 24a + 16) = 100
a² + 9a² + 4a + 24a + 4 + 16 = 100
10a² + 28a + 20 = 100
10a² + 28a = 100 - 20
10a² + 28a = 80
5a² + 14a = 40
5(a² + 14/5 a) = 40
a² + 2.8a = 40 / 5
a² + 2.8a = 8
a² + 2.8a = 8
a² + 2.8a + 1.96 = 8 + 1.96
(a + 1.4)² = 9.96
a + 1.4 = √9.96
a + 1.4 = ± 3.156
a = - 1.4 ± 3.156
If a = - 1.4 + 3.156,
a = 1.756
If a = - 1.4 - 3.156,
a = - 4.56
Feasible Domain: a > 0 | a in Q4
Since a is positive in Q4,
a = 1.756
and
- 3a = - 3(1.756)
- 3a = - 5.268
The point is (1.756, - 5.268)
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