Parabola 2 Question 15
15. Equation of common tangent of $y=x^{2}, y=-x^{2}+4 x-4$ is
$(2006,5 M)$
(a) $y=4(x-1)$
(b) $y=0$
(c) $y=-4(x-1)$
(d) $y=-30 x-50$
Passage Based Problems
Passage
Let $a, r, s, t$ be non-zero real numbers. Let $P\left(a t^{2}, 2 a t\right), Q, R\left(a r^{2}, 2 a r\right)$ and $S\left(a s^{2}, 2 a s\right)$ be distinct point on the parabola $y^{2}=4 a x$. Suppose that $P Q$ is the focal chord and lines $Q R$ and $P K$ are parallel, where $K$ is point $(2 a, 0)$.
(2014, Adv.)
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Solution:
- The equation of tangent to $y=x^{2}$, be $y=m x-\frac{m^{2}}{4}$. Putting in $y=-x^{2}+4 x-4$, we should only get one value of $x$ i.e. Discriminant must be zero.
$$ \begin{aligned} \therefore \quad m x-\frac{m^{2}}{4} & =-x^{2}+4 x-4 \\ \Rightarrow \quad x^{2}+x(m-4)+4-\frac{m^{2}}{4} & =0 \\ D & =0 \end{aligned} $$
Now, $\quad(m-4)^{2}-\left(16-m^{2}\right)=0$
$\Rightarrow 2 m(m-4)=0 \Rightarrow m=0,4$
$\therefore \quad y=0$ and $y=4(x-1)$ are the required tangents.
Hence, (a) and (b) are correct answers.
