Embibe Experts Solutions for Chapter: Ellipse, Exercise 3: Exercise-3

In a triangle$ABC$ with fixed base $BC,$ the vertex $A$ moves such that $\cos B+\cos C=4{\sin }^2\frac{A}{2}.$ If $a, b$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A, B$ and $C$, respectively, then

The normal at a point $P$ on the ellipse $x^2+4y^2=16$ meets the $x$-axis at $Q$. If $M$ is the mid-point of the line segment $P \text{and} Q$, then the locus of $M$ intersects the latus rectum of the given ellipse at the points.

Suppose that the foci of the ellipse $\frac{x^2}{9}+ \frac{y^2}{5}=1$ are $\left(f_1, 0\right) and (f_2, 0)$ where $f_1>0 and f_2<0.$ Let $P_1 and P_2$ be two parabolas with a common vertex at (0, 0) and with foci at $\left(f_1, 0\right)$ and $\left(2f_2, 0\right),$ respectively. Let $T_1$ be a tangent to $P_1$ which passes through $\left(2f_2, 0\right)$ and $T_2$ be a tangent to $P_2$ which passes through $(f_1, 0)$ . If $m_1$ is the slope of $T_1$ and $m_2$  is the slope of $T_2,$ then the value of $\left(\frac{1}{m_1^2}+ m_2^2\right)$ is

Answer $Q.39, Q.40$ and $Q.41$ by appropriately matching the information given in the three columns of the following table.

If the tangent to a suitable conic (List I) at $\left(\sqrt{3},\frac{1}{2}\right)$ is found to be $\sqrt{3}x+2y=4,$ then which of the following  options is the only CORRECT combination?

The ellipse $x^2+4y^2=4$ is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point $\left(4,0\right)$. Then, the equation of the ellipse is

An ellipse is drawn by taking a diameter of the circle ${\left(x-1\right)}^2+y^2=1$ as its semi-minor axis and a diameter of the circle $x^2+{\left(y-2\right)}^2=4$ as its semi-major axis. If the centre of the ellipse is the origin and its axes are the coordinate axes, then the equation of the ellipse is

The locus of the foot of perpendicular drawn from the centre of the ellipse $x^2+3y^2=6$ on any tangent to it is

The eccentricity of an ellipse whose centre is at the origin is $\frac{1}{2}.$  If one of its directrices is $x=-4,$ then the equation of the normal to it at $\left(1,\frac{3}{2}\right)$ is :