If a chord of the parabola y^2=4x passes thro | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let $P(t^2, 2t)$ be one end of a focal chord $PQ$ of the parabola $y^2=4x$. The coordinates of the other end $Q$ are $(\frac{1}{t^2}, \frac{-2}{t}

Vedclass · Accepted Answer

$4 \operatorname{cosec}^2 	heta$

Vedclass · Answer

(C) Let $P(t^2, 2t)$ be one end of a focal chord $PQ$ of the parabola $y^2=4x$. The coordinates of the other end $Q$ are $(\frac{1}{t^2}, \frac{-2}{t})$,where $tt' = -1$. Given that the chord makes an angle $	heta$ with the positive direction of the $X$-axis,the slope of the chord is $	an 	heta$. $	an 	heta = \frac{\frac{-2}{t} - 2t}{\frac{1}{t^2} - t^2} = \frac{-2(t + \frac{1}{t})}{\frac{1-t^4}{t^2}} = \frac{-2(t + \frac{1}{t})}{\frac{(1-t^2)(1+t^2)}{t^2}} = \frac{2t}{t^2-1}$. Alternatively,using the slope formula for a focal chord with parameter $t$: $	an 	heta = \frac{2}{t - \frac{1}{t}}$. Thus,$t - \frac{1}{t} = 2 \cot 	heta$. The length of the focal chord $PQ$ is given by $a(t + \frac{1}{t})^2$,where $a=1$. $PQ = (t + \frac{1}{t})^2 = (t - \frac{1}{t})^2 + 4$. Substituting $t - \frac{1}{t} = 2 \cot 	heta$: $PQ = (2 \cot 	heta)^2 + 4 = 4 \cot^2 	heta + 4 = 4(\cot^2 	heta + 1) = 4 \operatorname{cosec}^2 	heta$.

If a chord of the parabola $y^2=4x$ passes through its focus and makes an angle $\theta$ with the $X$-axis,then its length is

Similar Questions

Which one of the following is mainly responsible for the depletion of the ozone layer?

Which of the following statements is correct?

The monomer of neoprene is

$A$ tiny air bubble in a glass slab $(\mu = 1.5)$ appears from one side to be $6\, cm$ from the glass surface and from the other side,$4\, cm$. The thickness of the glass slab is......$cm$

$A$ current of $2 \, A$ flows in an electric circuit as shown in the figure. The potential difference $(V_R - V_S)$, in volts ($V_R$ and $V_S$ are potentials at $R$ and $S$ respectively) is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module