A circle of radius - r - is concentric with the ellipse x 2- a 2- y 2- b 2-1- Then inclination of common tangent with major axis is b - r - a A- tan -1-r2-b2-a2-r2B- tan -1b-aC- -2D- tan -1r b-a

The correct option is A The tangent of ellipse is y=mx+√(a^2m^2+b^2) this line touches x^2+y^2=r^2 Condition is | √(a^2m^2+b^2)/√(m^2+1) |=r a^2m^2+b^2=r^2m^2 ...

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Standard XII

Mathematics

Locus

A circle of r...

Question

A circle of radius ‘r’ is concentric with the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ .Then inclination of common tangent with major axis is _______(b<r<a)

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Solution

The correct option is C
The tangent of ellipse is $y = m x + \sqrt{a^{2} m^{2} + b^{2}}$ this line touches $x^{2} + y^{2} = r^{2}$
Condition is $∣ ∣ ∣ \frac{\sqrt{a^{2} m^{2} + b^{2}}}{\sqrt{m^{2} + 1}} ∣ ∣ ∣ = r$
$a^{2} m^{2} + b^{2} = r^{2} m^{2} + r^{2}$
$m^{2} (a^{2} - r^{2}) = r^{2} - b^{2} \Rightarrow m^{2} = \frac{r^{2} - b^{2}}{a^{2} - r^{2}}$
$m = \sqrt{\frac{r^{2} - b^{2}}{a^{2} - r^{2}}}$
Inclination is $t a n^{- 1} \sqrt{\frac{r^{2} - b^{2}}{a^{2} - r^{2}}}$

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A circle of radius ‘r’ is concentric with the ellipse x2a2+y2b2=1 .Then inclination of common tangent with major axis is _______(b<r<a)

The correct option is C The tangent of ellipse is y=mx+√a2m2+b2 this line touches x2+y2=r2 Condition is ∣∣∣√a2m2+b2√m2+1∣∣∣=r a2m2+b2=r2m2+r2 m2(a2−r2)=r2−b2⇒m2=r2−b2a2−r2 m=√r2−b2a2−r2 Inclination istan−1√r2−b2a2−r2

A circle of radius ‘r’ is concentric with the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ .Then inclination of common tangent with major axis is _______(b<r<a)