general geometric properties of parabola (1 Viewer)

[ozidolroks]

hey !

can someone please help me with this question

at point P(2ap,ap^2) on the parabola x^2=4ay a tangent is drawn to cut the axis of the parabola at T and the normal from P meets the axis in G.
If S is the focus:

b) if a line PN is drawn perpendicular to the zxis, meeting the axis in N, prove that the length of NG is a constant length for all positions of P.

3. Prove that if the tangent from a point P on a parabola meets the directrix at A , then the angle ASP is a right angle.

thankx

 

[Trebla]

hey !

can someone please help me with this question

at point P(2ap,ap^2) on the parabola x^2=4ay a tangent is drawn to cut the axis of the parabola at T and the normal from P meets the axis in G.
If S is the focus:

b) if a line PN is drawn perpendicular to the zxis, meeting the axis in N, prove that the length of NG is a constant length for all positions of P.

Equation of normal at P:
y - ap² = (-1/p)(x - 2ap)
Sub x = 0
y = 2a + ap²
.: G(0,2a + ap²)
Since PN is perpendicular to the axis of the parabola N is (0,ap²)
Length NG = 2a + ap² - ap²
= 2a
which is constant and hence independent of all positions of P

3. Prove that if the tangent from a point P on a parabola meets the directrix at A , then the angle ASP is a right angle.

thankx

Equation of tangent at P:
y - ap² = p(x - 2ap)
Sub y = - a
- a - ap² = px - 2ap²
x = a(p² - 1) / p
.: A(a(p² - 1) / p, - a)
m_AS = (a - (-a)) / (0 - a(p² - 1) / p)
= - 2p/(p² - 1)
m_PS = (ap² - a) / (2ap - 0)
= (p² - 1)/2p
m_AS x m_PS = (p² - 1)/2p x - 2p/(p² - 1)
= - 1
Hence angle ASP is a right angle.