#circles and lines

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devout grove
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bro can somone help explain these. thanks in advance

Screenshot_2566-11-12_at_21.23.10.png
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devout grove
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thanks

void onyx
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in one equation (usually the linear one) isolate one varaible

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then substitute it itno the other equation and solve for the remaining vairable

devout grove
wintry pivot
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What class is this for?

wintry pivot
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Ah. I looked up the underground mathematics website. It's UK A-levels. Makes sense. The questions are all at a pre-calculus level, but quite hard relative to what's done in American public schools.

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#4 Requires you to substitute mx + 1 for y in the equation of the circle, put the resulting quadratic equation in standard form, plug that into the quadratic formula and then determine what values of m make the discrimanant greater than 0.

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#7 is interesting too. Because the x-axis is tangent to the circles and the centers are on the line x + 2y = 22 the radius of each circle is just the distance from the center, which is on the line, to the x-axis. So you need to find two points on the line x + 2y = 22 such the the distance from the point to the x- axis is the same as the distance from the point to (8,2).

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Points on the line have the form (x, -(1/2)x + 11). The distance to the x-axis is the absolute value of the second coordinate and the distance to (8,2) can be found via the distance formula for points on the plane. Set the distances equal and you should get a quadratic equation with two solutions. Each solution will be the x value of the center of a circle. The y values can be gotten from the equation of the line. Those are your centers. The radius is just the distance to the x axis. Once you have the center and radius you can write the equation of a circle. For part B just show that the line 4x + 3y = 88 intersects each circle exactly once.

wintry pivot
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There's a solution to #7.

wintry pivot
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Here's a solution to #4)

m-values.png
devout grove
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4/6/7 right

wintry pivot
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You can read my handwriting? I did it on an e-ink tablet and uploaded it.

devout grove
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can u help with question 5 c and d

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i got up to a and b and got them correct

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but c and d couldnt figure it out

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thanks

devout grove