Question

1. given point a(3, -4) and b(8, -6) in the direction ab., what is the coordinate y of the point k that partitions ak of kb in the ratio 3:2.

a. - 2
b. 0
c. 2
d. 4

1. what is the equation of the circle with center at (-1, -3) and radius 4?

a. ((x + 1)^2 + (y + 3)^2 = 2)
b. ((x - 1)^2 + (y + 3)^2 = 2)
c. ((x + 1)^2 + (y - 3)^2 = 2)
d. ((x - 1)^2 + (y - 3)^2 = 2)

Explanation:

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Problem 1:

Step1: Recall section formula for y

The section formula for the y-coordinate of a point dividing a line segment joining $(x_1,y_1)$ and $(x_2,y_2)$ in ratio $m:n$ is $y = \frac{my_2 + ny_1}{m+n}$.

Step2: Substitute given values

Here, $A(3,-4)=(x_1,y_1)$, $B(8,-6)=(x_2,y_2)$, $m=3$, $n=2$.
$y = \frac{3(-6) + 2(-4)}{3+2}$

Step3: Calculate numerator and denominator

Numerator: $3(-6)+2(-4) = -18 -8 = -26$
Denominator: $3+2=5$
$y = \frac{-26}{5} = -5.2$
Note: There is a discrepancy with the provided options, but following the ratio $AK:KB=3:2$, this is the correct calculation. If the ratio was $KB:AK=3:2$, the result would be $y=\frac{2(-6)+3(-4)}{5}=\frac{-12-12}{5}=-4.8$, which also does not match options. If the question intended $AK:AB=3:2$, the point would lie outside the segment, $y=-4 + \frac{3}{2}(-6+4)=-7$, which also does not match.

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Problem 2:

Step1: Recall circle equation formula

The standard equation of a circle with center $(h,k)$ and radius $r$ is $(x-h)^2 + (y-k)^2 = r^2$.

Step2: Substitute given values

Center $(-1,-3)=(h,k)$, radius $r=4$.
$(x - (-1))^2 + (y - (-3))^2 = 4^2$

Step3: Simplify the equation

$(x+1)^2 + (y+3)^2 = 16$
Note: All provided options have an incorrect right-hand side (they use 2 instead of 16). The correct equation is not listed among the choices.

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Answer:

1. None of the provided options match the correct calculated value of $y=-5.2$.
2. None of the provided options are correct; the correct equation is $(x+1)^2 + (y+3)^2 = 16$.