Some more coordinate geometry questions (1 Viewer)

[Jonothancroller]

25. The line passing through M(a,b) intersects the x-axis at A and the y-axis at B. Find the equation of the line if: (a) M bisects AB, (b) M divides AB in the ratio 2 : 1, (c) M divides AB in the ratio k:l
26. The tangent to a circle is perpendicular to the radius at the point of contact. Use this fact to show that the tangent to x^2+y^2=r^2 at (a,b) is ax+by=r^2
27. Show that the parametric equations x= tcosθ + a and y= tsinθ + b represent a straight line through (a, b) with gradient m = tanθ
Next question is too big to write so here's a picture:
Thanks

 

[psyc1011]

27)
x = tcosθ + a -> cosθ = (x-a)/t
y = tsinθ + b -> sinθ = (y-b)/t

Divide to eliminate 't',

tanθ = (y-b)/(x-a)

Isolate y,

y = (x-a)tanθ + b

Test (a,b): b = (a-a)θ = b, works

So y = (x-a)m + b where m = tanθ as required

 

[calamebe]

I can do 26, 27, and 28 (a). I don't exactly know what they mean by perpendicular form of the line. Here are my answers: http://m.imgur.com/a/99DpH

 

[psyc1011]

26) To find the line, we need gradient and a point. We have the point, it is (a,b). We need gradient. We know that the tangent is perp. to the radius at the point of contact. So we find gradient of radius (0,0) to point of contact.

m(radius to point) = b/a

m(tangent) = -a/b

Point-gradient formual yields

y - b = -(a/b)(x-a)

by - b^2 = a^2 - ax

ax + by = a^2 + b^2 = r^2 since (a,b) lies on the circle so a^2 + b^2 = r^2

 

[Jonothancroller]

Does anyone else have answers for 25, and 28 b) and c)?
I also have this question: Show that every circle that passes through the intersections of the circle x^2 + y^2 = 2 and y=x can be written in the form (x-u)^2+ (y+u)^2 =2(1+u)^2

 

[FrankXie]

Does anyone else have answers for 25, and 28 b) and c)?
I also have this question: Show that every circle that passes through the intersections of the circle x^2 + y^2 = 2 and y=x can be written in the form (x-u)^2+ (y+u)^2 =2(1+u)^2

for question 25, let the equation of the line be y-b=m(x-a), then use the condition that M is the midpoint of A and B to for an equation in m. noting that a and b are constants, solve the equation you can find m, thus the equation of the line.

for the current one, following the idea of k-method used for finding equations of the line passing through the point of intersection of two given lines, we can deduce that any circle passing through the intersections of the circle x^2 + y^2 = 2 and and the line y=x can be written into the general form
( x^2 + y^2 - 2 ) + k ( x - y ) =0. starting from here, you can solve it.

 

[calamebe]

Does anyone else have answers for 25, and 28 b) and c)?
I also have this question: Show that every circle that passes through the intersections of the circle x^2 + y^2 = 2 and y=x can be written in the form (x-u)^2+ (y+u)^2 =2(1+u)^2

I've got your new question, though are you sure it isn't (x-u)^2+ (y+u)^2 =2(1+u^2)?
http://m.imgur.com/MmuoPz5

 

[Jonothancroller]

Thanks guys for the help and yes, I wrote the question wrong, sorry. Can anyone do 28 b) and c)?