Angle of Intersection of Two Curves

Q. If tangents are drawn from a point $P(2,0)$ to the curve $\sqrt{1+y^2}=\frac{x}{3}$ which meet the curve at $A$ and $B,$ then

1. acute angle between the tangents is ${\tan }^{-1}\left(\frac{\sqrt{5}}{2}\right)$
2. $AB=\sqrt{5}$
3. area of $△PAB$ is $\frac{5\sqrt{5}}{4}$ sq. units
4. $△PAB$ is equilateral

Q. If tangents are drawn from a point $P(2,0)$ to the curve $\sqrt{1+y^2}=\frac{x}{3}$ which meet the curve at $A$ and $B,$ then

1. acute angle between the tangents is ${\tan }^{-1}\left(\frac{\sqrt{5}}{2}\right)$
2. $AB=\sqrt{5}$
3. area of $△PAB$ is $\frac{5\sqrt{5}}{4}$ sq. units
4. $△PAB$ is equilateral

Q.

If the curves $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{{\alpha }^2}+\frac{y^2}{{\beta }^2}=1$ cut each other orthogonally,

1. $a^2+b^2={\alpha }^2+{\beta }^2$

2. $a^2-b^2={\alpha }^2-{\beta }^2$
   

3. $a^2-b^2={\alpha }^2+{\beta }^2$
   

4. $a^2+b^2={\alpha }^2-{\beta }^2$

Q. The acute angle between the curves $y^2=x\mathrm{and}{x}^2=y$ at (1, 1) is .

1. ${\tan }^{-1}\frac{4}{3}$
2. ${\tan }^{-1}\frac{3}{4}$
3. $90°$
4. $45°$

Q.

If the tan of angle of intersection of y = $x^2$ and y = $x^3$ in the first quadrant is m, find the value of $\left|\frac{1}{m}\right|$

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Q.

The angle of intersection of curves. y = [|sin x| + |cos x|] and $x^2+y^2=5$ where [.] denotes greatest integral function is

1. $\frac{\pi }{4}$

2. $ta{n}^{-1}\left(\frac{1}{2}\right)$

3. $ta{n}^1\left(2\right)$

4. none of these