4 Mixed Questions! (1 Viewer)

[kubekoo]

4 Mixed Questions!

Q1.
A is the point (-2,0) and B the point (2,0). A point P moves such that AP=3PB. Prove that the locus of P is a circle whose equation is x^2 + y^2 -5x +4 = 0 and find the centre and radius of this circle.

(This part already done, the following part is the part that I can’t do)

A point T moves such that the length of the tangent from T to the above circle is equal to the distance of T from the y axis. Find the equation of the locus of T.

Q2.
Find the length of the common chord of two intersection circles whose radii are 17cm and 10cm and whose centres are 21cm apart.

Q3.
Prove that the exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. X and Y are two points in the sides AB, AC of a triangle such that XBCY is cyclic. Prove that the tangent at A to the circumcircle of ABC must be parallel to XY.

Q4.
Express tan(2*theta) in terms of tan (theta).
A person observes the angles subtended by a tower and its spire to be equal. He knows the height h of the tower and his distance x from its foot. Prove that the height of the spire is: [(x^2 + h^2)*h]/(x^2 - h^2)

 

[Eagles]

can you get a diagram up for Q3, 4?

 

[Xayma]

Here is a diagram for Q 3 from the above information. Ill do 4 shortly.

Anyway for Q3. (After the showing of a cyclic quad angle thing)

∠ ABC= ∠ ACB (∠ between tangent and chord = ∠ in alternate segment)

∠ AXY= ∠ ACB (exterior angle of a cyclic quad = opposite internal ∠ )

∴ the tangent at A is parallel to XY

(∠ ABC=∠ AXY, alternate ∠'s = therefore the lines are parallel)

 

[Estel]

4.

tan2A = 2tanA/(1-tan^2A)

let H be height of spire

(H+h)/x = tan2A
H = (xtan2A-h)
but tan A = h/x
so H = x[(2h/x)/(1-h^2/x^2)] -h
= [2hx^2/x^2-h^2] -h
= [(2hx^2-h(x^2-h^2))/x^2-h^2]
=(x^2+h^2)h]/(x^2-h^2)