Question

List I		List II
P	If the distance between two parallel tangents having slope $m$ drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{49}=1$ is 2 , then $\frac{2}{5}|m|$ is equal to	1	5
Q	If a variable line has its intercepts on the coordinate axes e and e', where $\frac{e}{2}$ and $\frac{e^{\prime }}{2}$ are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle $x^2+y^2=r^2$, where $r$ is equal to	2	4
R	If the equation of line touching both parabolas $y^2-4x=0$ and $x^2+32y=0$ is $ax-2y+b=0(a,b\in R)$, then $(a+b)$ is equal to	3	2
S	If the mid-point of a chord of the ellipse $\frac{x^2}{16}+\frac{y^2}{25}=1$ is $M(0,3)$ and the length of chord is $\frac{8p}{5}$, then $p$ is equal to	4	1

• A. P= 4 ,Q= 3 ,R= 2 ,S= 1
• B. P= 3 ,Q= 1 ,R=2 ,S=4
• C. P=3 ,Q= 4 ,R= 1 ,S= 2
• D. P=4 ,Q= 3 ,R=1 ,S= 2

Answer 1

Answer: (D)

(P) The equation of tangents to the hyperbola having slope $m$ are $y=mx\pm \sqrt{9m^2-49}$.
Now, $\frac{2\sqrt{9m^2-49}}{\sqrt{1+m^2}}=2$ (Given)
$\Rightarrow m=\pm \frac{5}{2}\Rightarrow \frac{2}{5}|m|=1.$
(Q) $As,\frac{e}{2}$ and $\frac{e^{\prime }}{2}$ are eccentricities of a hyperbola and its conjugate hyperbola, so
$\frac{4}{e^2}+\frac{4}{e^{\prime 2}}=1\Rightarrow 4=\frac{e^2{e}^{\prime 2}}{e^2+e^{\prime 2}}$ ....(1)The equation of line is
$e^{\prime }x+ey-e{e}^{\prime }=0$
It is tangent to circle $x^2+y^2=r^2$
$\therefore \frac{e^{\prime }}{\sqrt{e^2+e^{\prime 2}}}=r=2\text{ [using(1)] }$
(R) The equation of common tangent is $x-2y+4=0$
$\therefore (a+b)=5$
(S) The equation of chord to ellipse whose midpoint is $(0,3)$ is $\left(T=S_1\right)$
$\Rightarrow y=3$. It intersects the ellipse
$\frac{x^2}{16}+\frac{y^2}{25}=1\Rightarrow \frac{x^2}{16}=1-\frac{9}{25}\Rightarrow x=\pm \frac{16}{5}\text{. }$
$\therefore$ Length of chord $=\frac{32}{5}\Rightarrow p=4$