Thread: Parametric tangents and normals

1. Parametric tangents and normals

P and Q are the points with parameters p and q on the parabola $x=2at , y=at^2$

If pq = 1 and S is the focus of the parabola, show that

1/SP + 1/SQ = 1/a

Thanks

2. Originally Posted by deltaxray
P and Q are the points with parameters p and q on the parabola $x=2at , y=at^2$

If pq = 1 and S is the focus of the parabola, show that

1/SP + 1/SQ = 1/a
The focus is at $S = (0,a)$. If $P = (2ap,ap^2)$ and $Q = (2aq,aq^2)$ then $SP^2 = 4a^2p^2 + a^2(1-p^2)^2 = a^2(1+p^2)^2$ and so $SP = a(1+p^2)$. Similarly $SQ = a(1+q^2)$.

Then $\frac1{SP} + \frac1{SQ} = \frac1{a(1+p^2)} + \frac1{a(1+q^2)} = \frac{2+p^2+q^2}{a(1+p^2)(1+q^2)}$. If $pq=1$ then you can write the numerator as $1+p^2+q^2+p^2q^2 = (1+p^2)(1+q^2)$, so the fraction simplifies to $\frac1a$.

3. Thank you