Q111 / 160 · TS EAMCET 2015 · Mathematics

For the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, a list of lines given in List I are to be matched with their equation given in List II. Figure
The correct matching isFigure

  • A
    a
  • Correct answer
    B
    b
  • C
    c
  • D
    d

Solution

Given equation of the ellipse is $$ \frac{x^2}{25}+\frac{y^2}{16}=1 $$ Here, $a=5$ and $b=4$ (P) Equation of the directrix corresponding to the focus $(-3,0)$ is $$ \begin{aligned} \quad x & =\frac{-a}{e} \\ \quad \text { Now, } \quad e & =\sqrt{1-\frac{b^2}{a^2}}=\sqrt{1-\frac{16}{25}}=\sqrt{\frac{9}{25}}=\frac{3}{5} \\ \therefore \quad \quad \quad \quad \quad \quad & =-\frac{5}{3 / 5} \\ \Rightarrow \quad 3 x & =-25 \Rightarrow 3 x+25=0 \end{aligned} $$ (Q) Equation of tangent at the vertex $(0,4)$ is $$ y=b \Rightarrow y=4 $$ (R) Equation of latusrectum through $(3,0)$ is $$ \begin{aligned} x & =a e \\ x & =5 \times \frac{3}{5} \Rightarrow x=3 \end{aligned} $$

Hence, the correct answer is option: 'B'.

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