Question

The equations of the tangents to the ellipse \(\frac{x^2}{16}+\frac{y^2}{9} = 1\) making an inclination of \(30^{\circ}\) with the major axis are

• A. \(x+\sqrt{3}\,y\pm \sqrt{43} = 0\)
• B. \(x-\sqrt{3}\,y\pm \sqrt{43} = 0\)
• C. \(\sqrt{3}\,x-y\pm \sqrt{43} = 0\)
• D. \(x-\sqrt{3}\,y\pm \sqrt{3} = 0\)

Hint:

Use \(y=mx\pm\sqrt{a^2m^2+b^2}\) with \(m=\tan30^{\circ}\).

Answer 1

The Correct Option is B

Step 1: Use the condition for tangency
A line $y=mx+c$ touches the ellipse when $c^2=a^2m^2+b^2$.

Step 2: Apply
With $m=1/\sqrt3$, $c^2=\frac{16}{3}+9=\frac{43}{3}$, $c=\pm\sqrt{43/3}$. Rearranging $y=x/\sqrt3+c$ gives $x-\sqrt3y\pm\sqrt{43}=0$, option (B).

Final Answer:
Option (B) gives the tangents. \[ \boxed{x-\sqrt3\,y\pm\sqrt{43}=0} \]