Question

The length of a common tangent to the curves ${4x}^2+{25y}^2=100$ and $x^2+y^2=16$ intercepted by the coordinate axes is

• A. 7 th term of sequence $\frac{2\sqrt{3}}{3},\frac{4\sqrt{3}}{3},\frac{6\sqrt{3}}{3}$,......
• B. $\frac{14}{\sqrt{3}}$
• C. $\frac{7}{\sqrt{3}}$
• D. 7 th term of sequence $\frac{1}{\sqrt{3}},\frac{2}{\sqrt{3}},\frac{4}{\sqrt{3}}$,......

Answer 1

Answer: (B)

$\frac{14}{\sqrt{3}}$

$\frac{x^2}{25}+\frac{y^2}{4}=1$

$x^2+y^2=16$

Let$y=mx+\sqrt{a^2{m}^2+b^2}$ be a tangent to the ellipse

$⟹y=mx+\sqrt{25m^2+4}$

Let $y=mx+r\sqrt{4m^2+1}$ be a tangent to the circle

$⟹y=mx+4\sqrt{m^2+1}$

Both equations must represent the same line for it to be a common tangent

$4\sqrt{m^2+1}=\sqrt{25m^2+4}$

$⟹16m^2+16=25m^2+4$

$9m^2=12$

$m=\frac{2}{\sqrt{3}}$

The equation is

$y=\frac{2}{\sqrt{3}}x+4\sqrt{\frac{4}{3}+1}$

$\sqrt{3}y=2x+4\sqrt{7}$

$y=0,x=-2\sqrt{7}⟹A(-2\sqrt{7},0)$

$x=0,y=\frac{4\sqrt{7}}{\sqrt{3}}⟹B(0,\frac{4\sqrt{7}}{\sqrt{3}})$

$Distance=l=\sqrt{\frac{4\times 7+16\times 7}{3}}=\sqrt{\frac{28\times 7}{3}}=\frac{14}{\sqrt{3}}$

$\frac{2\sqrt{3}}{3},\frac{4\sqrt{3}}{3},\frac{6\sqrt{3}}{3}\dots .$ is in AP

With $a=\frac{2\sqrt{3}}{3},d=\frac{2\sqrt{3}}{3}$

$a_7=a+6d=\frac{2\sqrt{3}}{3}+\frac{12\sqrt{3}}{3}=\frac{2\sqrt{3}}{3}=\frac{14}{\sqrt{3}}$