A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with
Select the correct option:
A
length of latus rectum 3
B
length of latus rectum 6
C
focus $\left(\frac{4}{3}, 0\right)$
D
focus $\left(0, \frac{3}{4}\right)$
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
Let Point $\mathrm{P}(\mathrm{x}, \mathrm{y})$
$$
\begin{aligned}
& Y-y=y^{\prime}(X-x) \\
& Y=0 \Rightarrow X=x-\frac{y}{y^{\prime}} \\
& Q\left(x-\frac{y}{y^{\prime}}, 0\right)
\end{aligned}
$$
Mid Point of PQ lies on axis
$$
\begin{aligned}
& x-\frac{y}{y^{\prime}}+x=0 \\
& y^{\prime}=\frac{y}{2 x} \Rightarrow 2 \frac{d y}{y}=\frac{d x}{x} \\
& 2 \ell n y=\ell n x+\ell n k \\
& y^2=k x
\end{aligned}
$$
It passes through $(3,3) \Rightarrow \mathrm{k}=3$
curve $\mathrm{c} \Rightarrow \mathrm{y}^2=3 \mathrm{x}$
Length of L.R. $=3$
$$
\text { Focus }=\left(\frac{3}{4}, 0\right)
$$
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