- Home
- Standard 11
- Mathematics
Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E: \frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P: y^2=12 x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the point $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?
($A$) The area of the quadrilateral $A_1 A _2 A _3 A _4$ is $35$ square units
($B$) The area of the quadrilateral $A_1 A_2 A_3 A_4$ is $36$ square units
($C$) The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-3,0)$
($D$) The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-6,0)$
$A,C$
$A,D$
$B,C$
$B,D$
Solution
$y=m x+\frac{3}{m}$
$C^2=a^2 m^2+b^2$
$\frac{9}{m^2}=6 m^2+3 \quad \Rightarrow m^2=1$
$\mathrm{T}_1 \& \mathrm{~T}_2$
$y=x+3, y=-x-3$
Cuts $\mathrm{x}$-axis at $(-3,0)$
$\mathrm{A}_1(3,6) \mathrm{A}_4(3,-6)$
$\mathrm{A}_2(-2,1) \mathrm{A}_3(-2,-1)$
$\mathrm{A}_1 \mathrm{~A}_4=12, \quad \mathrm{~A}_2 \mathrm{~A}_3=2, \quad \mathrm{MN}=5$
$\text { Area }=\frac{1}{2}(12+2) \times 5=35 \text { sq.unit }$
Ans. $(A, C)$
