Question

If \(\theta\) is acute angle between tangents drawn from point \((3,4)\) to ellipse \[ \frac{x^2}{25}+\frac{y^2}{9}=1 \] then \(\theta=\)

• A. \(Tan^{-1}\left(\frac{16}{9}\right)\)
• B. \(Tan^{-1}\left(\frac{32}{9}\right)\)
• C. \(Tan^{-1}\left(\frac9{25}\right)\)
• D. \(Tan^{-1}\left(\frac{16}{25}\right)\)

Hint:

For tangent angle questions from external point to conics, remember director circle shortcuts.

Answer 1

The Correct Option is B

Concept: Angle between tangents from external point to ellipse obtained through director circle relation. For ellipse: \[ \frac{x^2}{25}+\frac{y^2}{9}=1 \] director circle: \[ x^2+y^2=34 \]

Step 1: Distance from point to center.
\[ d=\sqrt{3^2+4^2}=5 \]

Step 2: Apply tangent angle relation.
Using standard formula for ellipse tangent pair angle: \[ \tan\theta=\frac{32}{9} \] Hence \[ \boxed{Tan^{-1}\left(\frac{32}{9}\right)} \]