Position of a Line W.R.T Ellipse

Q. If the tangent to the ellipse $x^2+4y^2=16$ at the point $P\left(\varphi \right)$ is normal to the circle $x^2+y^2-8x-4y=0,$ then possible values(s) of $\varphi$ is/are

1. $\frac{\pi }{4}$
2. $0$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{2}$

Q.

Find the set of values of $\lambda$ for which the
line $3x-4y+\lambda =0$ intersects the ellipse
$\frac{x^2}{16}+\frac{y^2}{a}=1$ at 2 distinct point.

1. $(-12,12)$

2. $(-12\sqrt{2},12\sqrt{2})$

3. $(12,12\sqrt{2})$

4. $(-12,12\sqrt{2})$

Q. The value(s) of $\lambda$ for which the line $y=x+\lambda$ touches the ellipse
$9x^2+16y^2=144$ is/are:

1. $5$
2. $-5$
3. $10$
4. $-10$

Q. For the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ and circle $(x-1{)}^2+(y+2{)}^2=3,$ the centre of the circle lies

1. inside the ellipse
2. outside the ellipse
3. on the ellipse
4. none of these

Q. If the latus-rectum through one focus of a hyperbola subtends a right angle at the farther vertex, then write the eccentricity of the hyperbola.