HELP!!! 1 Paramter Question and Few Circle Geom! (1 Viewer)

[kubekoo]

HELP!!! 1 Paramter Question and Few Circle Geom!

Q:
[This is come from 3Units Jones and Couchman Book2 Ex23.6 Q15b]
PQ is a focal chord in the parabola x^2 = 4ay. (x^2 is x squared)
PT is drawn parallel to the tangent at Q and QT is drawn parallel to the tangent at P. Show that the locus of T is x^2 = a (y-3a)
(I done this part already, the following part is the part that I can’t do)
If M is the midpoint of the focal chord PQ and a line through M, parallel to the axis of the parabola, meets the normal at P in A, find the locus of A.

[The following are Circle Geom question, come from 3units Fitzpatrick]
[Ch 20 Ex 20c, Q16,18, 23, 26, 28, 29-35]
[Note: Q29 i didn't type it up, since it has a diagram with it, and that is one of my problem as well, so if you know how to do it, please help me too!! thx!!!]
1. Calculate the distance of a chord of length 24 cm from the centre of a circle of radius 13cm. (I already done this section, the following section is the part I can’t do)
Also calculate the length of the tangents drawn from an external point to the extremities of the chord

2. If the radii of two intersecting circles are 17cm and 10cm and the length of the common chord is 16cm, calculate the length of the line joining the centres of the circles and the length of the common tangent.

3. Two circles intersect at A and B. The tangent to the first circle at A cuts the second circle at C and the tangent to the second circle at A cuts the first circle at D. Prove that triangle ABC and triangle DBA are similar.

4. Two circles intersect at A and B. The tangent to the second circle at A cuts the first circle at C and the tangent to the first circle at B cuts the second circle at D. Prove that AD is parallel to BC.

5. Two circles touch internally at A. The tangent at P on the smaller circle cuts the larger circle at Q and R. Prove that AP bisects angle RAQ.

6. Two circles, centres O and P, touch externally at A. The direct common tangent touches the circles at X and Y respectively. The common tangent at A meets the direct common tangent at Z. Prove that ZX = ZY, and angle OZP = 90degree

7. Two circles intersect in X and Y. The tangent at X to the first circle cuts the second circle at A and AY produced cuts the first circle at B. Prove that XB is parallel to the tangent at A to the second circle.

8. ABC is a triangle inscribed in a circle. The tangent at C meets AB produced at P and the bisector of angle ACB meets AB at Q. Prove that PC = PQ

9. PA and PB are two tangents to a circle and X is the midpoint of the minor arc AB. Prove that XA bisects angle PAB, and XB bisects angle PBA.

10. Two circles intersect at A and B. A line through A cuts the first circle at P and the second circle at Q. From an external point T a tangent TP is drawn and TQ produced meets the second circle again at R. Prove that the points P, T, R, B are concyclic.

Looking forward to hear from any one of you! Thx thx!!!

 

[withoutaface]

1. Use SOHCAHTOA to find the size of angle TOB within triangle OXB=arcsin(12/13)

Then use it again to within triangle TOB to find TB= tan(arcsin[12/13])/13

Now TB=TA (tangents from an external point are equal)

 

[ngai]

here we go:

parabola Q:
x coords of M: x=1/2 (2ap+2aq) = a(p+q)
axis is vertical, so the line parallel to axis is: x=a(p+q)
normal at P is py+x=2ap+ap^3
sub x=a(p+q) into normal eqn:
py + ap + aq = 2ap + ap^3
y = (ap+ap^3-aq)/p
= a + ap^2 - aq/p
now, from first part, im sure u got pq = -1, so p = -1/q, and so q/p = -q^2
therefore y = a + ap^2 + aq^2
and (x/a) = p+q, so y = a + a(p^2 + 2pq + q^2 - 2pq)
y = a + a[(x/a)^2 - 2(-1)]
simplifying to x^2 = a(y-3a)

Q1. first part u did, and i got distance is 5
2nd part: draw in the two tangents, the radii that meet the tangents, and the line between the centre and the tangent intersections

u can show using congruent triangles:
the chord is perpendicular to the line betewen centre and tangent intersection
the two angles at the centre are the same, and both are equal to arctan(12/5)

since tangent perpendicular to radius, then the two angles at the tangent intersection are each 90-arctan(12/5)
then, if L is the length of tangent:
Lsin[90-arctan(12/5)] = 12
so L = 12/cos(arctan(12/5)) = 12/cos(arccos(5/13)) = 12/(5/13) = 156/5

Q2. that chord is perpendicular to line joining centres...prove by congruent triangles or wateva
radius perpendicular to chord bisects chord, so now ur chord has two lengths of 8
and ur radius is 10 and 17, so length of line between centres is sqrt(100-64) + sqrt(289-64) = 6 + 15 = 21

let length of common tangent be d
draw in the radii to meet the tangent
tangent is perpendicular to radius
from centre of small circle, create line parallel to the common tangent to meet the radius on big circle
clearly this line has length d
then u see theres a rt angled triangle, so d^2 + (R - r)^2 = 21^2, where R = 17, r = 10
so d = sqrt392 = 14rt2

Q3. angle ADB = angle BAC (angle in alt segment)
similarly, ang BAD = ang BCA
so similar triangles (equiangular)

Q4. ang CAB = angle between tangent-at-B and CB (ang in alt segment)
and similarly, angCAB = angADB
so angCAB = angADB and hence parallel lines (corresp. angles)

Q5. let QA meet little circle at C, and RA meet little circle at B
join BC
let angle between tangent-at-A and chord AC be x
so angCPA = x, and angQRA=x (angle in alt segment)
let angle between tangent-at-A and RA be y
so angRQA = y and angBPA = y (similarly)

angCBA = x (angle at circumference standing on same arc for little circle)
so CB // QR (corresp angles equal)
so angRPB = angPBC (alternate angles)
and angPBC = angPAC (angle at circumference ...)
also, angRPB = angBAP (alternet segment)
so angPAC = angBAP (= angRPB)


Q6. ZX = ZA (tangents to a external pt are equal)
similarly ZY = ZA
so ZX = ZY

by congruent triangles (SSS), u get angZOA = angZOX and angZPO = angZPY
Let angZOA = a, angZPO = b
then in quadrilateral XYPO, 2a + 2b + 2*90 = 360 (angle sum of quad, and radius perpendicular to tangent)
hence a+b = 90, and angOZP = 180 - (a+b) = 90

Q7. let angle between tangent-at-A and YA be x
then angYXA = x (alt segment)
and angXBY = x (similarly)
so lines are parallel (alt angles equal)

Q8. let angBCQ = angACQ = x
let angPCB = y, then angPCB = angCAB = y (alt segment)
clearly angPCQ = x+y
and angPQC = a+b (exterior ang of triangle equal to sum of interior opposite)
so PC = PQ (= angles opposite = sides)

Q9. equal arcs subtend equal chords, and so angXAB = angXBA = x
therefore angPAX = angXBA = x (alt segment)
similarly, angPBX = angBAX = x
so all those little angles are = x, and so those lines bisect the angles

Q10. let angle between tangnet-at-P and PB be x
so angPAB = x (alt segment again)
and angQRB = x (exterior angle of cyclic AQRB equal to interior opposite)
so PRTB cyclic (since exterior angle is equal to interior opposite)



Edit: stupid code stuffs up my post wen i use the less than symbol to denote 'angle'

 

[ngai]

oh and also, TEN geometry questions isnt exactly wat i call a "few"

 

[steverulz55]

guess you left out Q29 ngai

join up centres of circles (*)
join from centre of top circle to meet line joining bottom circle's centres => it is perpendicular as the lines in (*) form a equilateral triangle
altitude of triangle = rt ((2+2)^2 - 2^2) = rt12 (by pythagoras)

therefore P to AB = rt 10 + 2x radius of circle = rt 12 + 4 = 4 + 2rt3

 

[kubekoo]

thx thx thx!~ ty sooooo much~

but 2 of them i m bit lost~ but anyways~ when i read over it more than once, then i should be fine!! thx thx!!!!

 

[kubekoo]

Q10. let angle between tangnet-at-P and PB be x
so angPAB = x (alt segment again)
and angQRB = x (exterior angle of cyclic AQRB equal to interior opposite)
so PRTB cyclic (since exterior angle is equal to interior opposite)

denote 'angle'

10. Two circles intersect at A and B. A line through A cuts the first circle at P and the second circle at Q. From an external point T a tangent TP is drawn and TQ produced meets the second circle again at R. Prove that the points P, T, R, B are concyclic.

For this quesiton, i don't get ur solutino at all!! ><~
can you try to explain it again?! plz?! becos the logic doesnt' wound alright to me!! ><~

 

[Heinz]

10. Two circles intersect at A and B. A line through A cuts the first circle at P and the second circle at Q. From an external point T a tangent TP is drawn and TQ produced meets the second circle again at R. Prove that the points P, T, R, B are concyclic.

For this quesiton, i don't get ur solutino at all!! ><~
can you try to explain it again?! plz?! becos the logic doesnt' wound alright to me!! ><~

Check out the attached diagram. Youll get it

 

[kubekoo]

omg!!!

Got it now!!! ty!!! these question are just too hard for me!! ><