Question

Consider the ellipse $(3x-6{)}^2+(3y-9{)}^2=\frac{4}{169}(5x+12y+6{)}^2$

Column I contains the distances associated with this ellipse and Column II gives their value. Match the expressions/statements in column I with those in column II.

$\begin{array}{cccc}& \text{Column -I}& & \text{Column -II}\\ \left(a\right)& \text{The length of major axis}& \left(p\right)& \frac{72}{5}\\ \left(b\right)& \text{The length of minor axis}& \left(q\right)& \frac{16}{\sqrt{5}}\\ \left(c\right)& \text{The length of latus recturn}& \left(r\right)& \frac{16}{3}\\ \left(d\right)& \text{The distance between the directrices}& \left(s\right)& \frac{48}{5}\end{array}$

• A. A-S, B-Q, C-R, D-P
• B. A-Q, B-S, C-R, D-P
• C. A-R, B-P, C-S, D-P
• D. A-S, B-P, C-Q, D-S

Answer 1

The correct option is A

A-S, B-Q, C-R, D-P

Rewrite the equation as $(x-2{)}^2+(y-3{)}^2=\frac{4}{9}{\left[\frac{5x+12y+6}{13}\right]}^2$

$d=\frac{5.2+12.3+6}{\sqrt{5^2+{12}^2}}=\frac{52}{13}=4$

Also $e=\frac{2}{3}$

$d=(ae+h)-(\frac{a}{e}+h)\Rightarrow a=\frac{24}{5}$

Length of major axis $=2a=\frac{48}{5}$

Length of minor axis = (Length of major axis) $\sqrt{1-e^2}=\frac{16}{\sqrt{5}}$

Length of latusrectum = (Length of minor axis) $\sqrt{1-e^2}=\frac{16}{3}$

Distance between the directrices = (Length of major axis) $\times \frac{1}{e}=\frac{72}{5}$