Parabola Ques 7
- Let $A$ and $B$ be two distinct points on the parabola $y^2=4 x$. If the axis of the parabola touches a circle of radius $r$ having $A B$ as its diameter, then the slope of the line joining $A$ and $B$ can be
(2010)
(a) $-\frac{1}{r}$
(b) $\frac{1}{r}$
(c) $\frac{2}{r}$
(d) $-\frac{2}{r}$
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Answer:
Correct Answer: 7.(c,d)
Solution: (c,d) Here, coordinate $M=\left(\frac{t_1^2+t_2^2}{2}, t_1+t_2\right)$ i.e. mid-point of chord $A B$.
$ M P=t_1+t_2=r $ $\quad$ ……..(i)
Also, $\quad m_{A B}=\frac{2 t_2-2 t_1}{t_2^2-t_1^2}=\frac{2}{t_2+t_1}\quad\quad$ [when $A B$ is chord]
$\Rightarrow \quad m_{A B}=\frac{2}{r}\quad\quad$ [from Eq. (i)]
Also, $\quad m_{A^{\prime} B^{\prime}}=-\frac{2}{r}\quad\quad$ [when $A^{\prime} B^{\prime}$ is chord]
Hence, (c, d) are the correct options.
BETA
