Parabola and Ellipse — NDA Mathematics PYQs

Directions : For the next two (02) items that follow :
The foci of the ellipse $p{x}^2+16y^2=16p$ and the foci of the hyperbola $25(81x^2-144y^2)=11664$ coincide (assume $p<16$).

What is the value of p ?

Directions : For the next two (02) items that follow :
The foci of the ellipse $p{x}^2+16y^2=16p$ and the foci of the hyperbola $25(81x^2-144y^2)=11664$ coincide (assume $p<16$).

What is the difference between the eccentricities of the hyperbola and the ellipse ?

A tangent to the parabola $y^2=4x$ is inclined at an angle $45°$ with the positive direction of $x$-axis. What is the point of contact of the tangent and the parabola?

If any point on an ellipse is $(3\sin \alpha ,5\cos \alpha )$, then what is the eccentricity of the ellipse?

Consider the following statements in respect of the equation $x^2+3y=0$ :
I. The equation represents the equation to parabola that opens upwards.
II. The axis of the parabola is $x=0$.
III. The equation of the latus rectum is $4y-3=0$.
How many of the statements given above are correct?

In the parabola $y^2=8x$, the focal distance of a point $P$ lying on it is 8 units. Which of the following statements is/are correct ?

1. The coordinates of $P$ can be $(6,4\sqrt{3})$.
2. The perpendicular distance of $P$ from the directrix of parabola is 8 units.

Select the correct answer using the code given below :

What is the eccentricity of the ellipse if the angle between the straight lines joining the foci to an extremity of the minor axis is $90^{\circ }$ ?

The foci of the ellipse $4x^2+9y^2=1$ are at $Q$ and $R$. If $P(x,y)$ is any point on the ellipse, then what is $PQ+PR$ equal to?

Consider the points $P(4k,4k)$ and $Q(4k,-4k)$ lying on the parabola $y^2=4kx$. If the vertex is $A$, then what is $\angle PAQ$ equal to?

*Directions : Consider the following for the next two (02) items that follow :

$P(x,y)$ is any point on the ellipse $x^2+4y^2=1$. Let $E,F$ be the foci of the ellipse.*

What is $PE+PF$ equal to ?