Question

The number of focal chord(s) of length 4/7 in the parabola $7y^2=8x$ is

• A. 1
• B. 0
• C. infinite
• D. none of these

Answer 1

The correct option is B 0

The number of focal chord of length $\frac{4}{7}$ in parabola $7y^2=8x$ is

Length of focal chord PQ where $P({at}^2,2at)$ and $Q(\frac{a}{t^2},\frac{-2a}{t})$

$\sqrt{{(\frac{a}{t^2}+{at}^2)}^2+{(\frac{-2a}{t}-2at)}^2}a\sqrt{{(\frac{1}{t^2}+t^2)}^2+4{(\frac{1}{t}+t)}^2}{y}^2=\frac{8}{7}xa=\frac{2}{7}$

According to Q

$\frac{2}{7}\sqrt{{(\frac{1}{t^2}-t^2)}^2+4{(\frac{1}{t}+t)}^2}=\frac{4}{7}{(\frac{1}{t^2}-t^2)}^2+4{(\frac{1}{t}+t)}^2=4{(\frac{1}{t^2}+t^2)}^2[{(\frac{1}{t}-t)}^2+4]=4{(\frac{1}{t}+t)}^3=4\frac{1}{t}+t=4^{\frac{1}{3}}{t}^2-4^{\frac{1}{3}}t+1=0t=\frac{4^{\frac{1}{3}}\pm \sqrt{4^{\frac{2}{3}}-4}}{2}$

No real value

Therefore , there is zero focal chord