Answer:
Step-by-step explanation:
we know that
Triangles MNP and MAB are similar by AAA theorem
If two figures are similar, then the ratio of its corresponding sides is equal
so
substitute the values
What are the solutions to the system of equations? {y=2x2+6x−10 y=−x+5
2 answers:
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Answer:
6. C: {x^2 +(y-1)^2 =2; x+y = 3}
15. C: The line does not intersect the circle.
Step-by-step explanation:
The formula for the distance (d) from a point (x, y) to a line ax+by=c is ...
d = |ax+by-c|/√(a^2+b^2)
The formula for a circle centered at (h, k) with radius r is ...
(x -h)^2 +(y -k)^2 = r^2
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6. Comparing the circle equation to the generic equation, we find (h, k) = (0, 1) and r = √2. Then we want to find the line that is distance √2 from the center of the circle. Our line equation is x+y=c for some value of c that we want to find.
d = √2 = |0 +1 -c|/√(1^2+1^2)
2 = |1-c|
±2 = 1-c
c = 1±2 = -1 or 3
The line that is tangent to the circle is the one of choice C: x+y = 3
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The attached graph shows the lines for all 4 answer choices. The point of tangency is (1, 2), so x+y=1+2=3.
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15. The circle is centered at (4, 1) and has radius 3. The distance from the circle center to the line is ...
d = |2(4) -(1)|/√(2^2+(-1)^2) = 7/√5 ≈ 3.13
The distance from the circle center to the line is more than the radius of the circle, so there can be no points of intersection.
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Alternate solution
You can substitute for y using the equation of the line. Then the circle equation becomes ...
(x -4)^2 + (2x -1)^2 = 9
x^2 -8x +16 +4x^2 -4x +1 = 9
5x^2 -12x +8 = 0
The discriminant of this quadratic is ...
b^2 -4ac = (-12)^2 -4(5)(8) = 144-160 = -16
Since this value is negative, there can be no real solutions, meaning the line does not intersect the circle.
Answer:
Step-by-step explanation: