Question

​​​​​$\begin{array}{cccc}& \text{List 1}& & \text{List II}\\ \text{(1)}& \text{If PQ is a focal chord of the ellipse}& & \\ & \frac{x^2}{25}+\frac{y^2}{16}=1\text{which passes through S(3,0) \&}& & \\ & \text{PS=2, then the length of the focal chord PQ is}& \left(P\right)& 1\\ \text{(2)}& \text{The focal chord of}{y}^2=16x\text{is tangent to}& & \\ & (x-6{)}^2+y^2=2\text{. Then the possible values}& & \\ & \text{of the slopes of the chord is}& \left(Q\right)& 3\\ \text{(3)}& \text{A circle is described on the focal chord of}& & \\ & y^2=-12x\text{as a diameter such that it touches}& & \\ & \text{the line}x=a\text{. Then the value of}a\text{is}& \left(R\right)& 6\\ \text{(4)}& \text{If the eccentricity of the hyperbola is}\frac{5}{4}and& & \\ & \text{2x+3y-10=0 is a focal chord of the hyperbola}\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\text{,}& & \\ & \text{then the length of transverse axis is}& \left(S\right)& 8\\ & & \left(T\right)& 10\\ & & & \\ & & \left(U\right)& 4\end{array}$
Which of the following is only correct combination?

• A. $1\to \left(P\right)$
• B. $2\to \left(Q\right)$
• C. $3\to \left(R\right)$
• D. $4\to \left(S\right)$

Answer 1

The correct option is D $4\to \left(S\right)$

1)
Semi-latus rectum is the harmonic mean of the segment of focal chord
For $\frac{x^2}{25}+\frac{y^2}{16}=0$, let $PQ$ be the focal chord & $S$ be the focus
$\frac{2a}{b^2}=\frac{1}{SP}+\frac{1}{SQ}\Rightarrow SQ=8\Rightarrow PQ=10$

2)
In $\Delta SPC$,$PC=\sqrt{2},CS=2$
$\sin \theta =\frac{CP}{CS}=\frac{1}{\sqrt{2}}$
$\tan \theta =m=1$

3)
A circle described on the focal chord as diameter touches the directrix
For $y^2=-12x,x=3$ is the directrix

4)
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,e=\frac{5}{4}$ & $2x+3y-10=0$
passes through $(ae,0)$.
$2ae=102a\frac{5}{4}=10\Rightarrow a=42a=8$