## Ellipse Questions and Answers Part-3

1. $P\left(x_{1},y_{1}\right)$    and $Q\left(x_{2},y_{2}\right)$    , $y_{1} < 0 ,y_{2} < 0$    be the end points of the latus rectum of the ellipse $x^{2}+4y^{2}=4$    the equations of the parabolas with latus rectum PQ are
a) $x^{2}+2\sqrt{3y}=3+\sqrt{3}$
b) $x^{2}-2\sqrt{3y}=3+\sqrt{3}$
c) $x^{2}+2\sqrt{3y}=3-\sqrt{3}$
d) Both b and c

Explanation: Eccentricity e of the ellipse is given by

2. In a $\triangle ABC$   with fixed base BC, the vertex A moves such that $\cos B+\cos C = 4\sin^{2}\left(A/2\right)$
If a, b and c denote the sides of the triangle opposite to the angles A, B and C respectively, then
a) b + c = 4a
b) b + c = 2a
c) locus of point A is an ellipse
d) Both b and c

Explanation:

3. $E_{1}:x^{2}+2y^{2}-6x-12y+23=0$
and $E_{2}:4x^{2}+2y^{2}-20x-12y+35=0$
are two ellipse. The points of intersection of $E_{1}$  and $E_{2}$  lie on a circle with
a) centre at $\left(\frac{8}{3},3\right)$
b) centre at $\left(-\frac{8}{3},3\right)$
c) radius equal to $\frac{1}{3}\sqrt{\frac{47}{2}}$
d) Both a and c

Explanation: Equation of any curve passing through the

4. If the normal at any point P on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$    meets the major axis at $G_{1}$  and the minor axis at $G_{2}$  , then
a) $PG_{1}=\frac{b}{a}\sqrt{b^{2}\cos^{2}\theta+a^{2}\sin^{2}\theta}$
b) $PG_{2}=\frac{a}{b}\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}$
c) $PG_{1}:PG_{2}=b^{2}:a^{2}$
d) Both a and c

Explanation: Let the coordinates of P be (a cos $\theta$ , b sin $\theta$ )

5. Let $E_{1}$  and $E_{2}$  be two ellipses whose centres are at the origin. The major axes of $E_{1}$  and $E_{2}$  lie along the x-axis and the y-axis, respectively. Let S be the circle $x^{2}+\left(y-1\right)^{2}=2.$
The straight line x + y = 3 touches the curves S, $E_{1}$  and $E_{2}$  at P, Q and R, respectively. Suppose that $PQ=PR=\frac{2\sqrt{2}}{3}.$
If $e_{1}$  and $e_{2}$  are the eccentricities of $E_{1}$  and $E_{2}$  , respectively, then the correct expression(s) is(are)
a) $e_1^2+e_2^2=\frac{43}{40}$
b) $e_1+e_2=\frac{\sqrt{7}}{2\sqrt{10}}$
c) $\mid e_1^2-e_2^2\mid=\frac{5}{8}$
d) Both a and b

Explanation: P is the point of intersection of the tangent x + y = 3 to S and normal to S at P, that is, of (x – 0) – (y – 1) = 0. Thus, coordinates of P are (1, 2)

6. Let $F_{1}$  and $F_{2}$  be the foci of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$     and (0, b) be an end point of the minor axis. If triangle $BF_1F_2$  is equilateral, e is the eccentricity of the ellipse and $\triangle$  is the area of the triangle $BF_1F_2$  , then
a) $e=\frac{1}{2}$
b) $e=\frac{1}{3}$
c) $\triangle=\frac{\sqrt{3}}{4}a^{2}$
d) Both a and c

Explanation: Coordinates of F1 are (ae, 0) and of F2 are (–ae, 0)

7. If $\frac{x^{2}}{t^{2}-5t+6}+\frac{y^{2}}{5-4t-t^{2}}=1$       represents an ellipse but not a circle, then possible values (s) of t is (are)
a) $\frac{-\left(\sqrt{5}+1\right)}{4}$
b) $\frac{-\left(\sqrt{3}+1\right)}{3}$
c) $\frac{13}{16}$
d) All of the Above

Explanation:

8. Let $e\left(\lambda\right)$   be the eccentricity of the ellipse $\frac{x^{2}}{a^{2}+\lambda}+\frac{y^{2}}{b^{2}+\lambda}=1$     , where $a>b, \lambda\geq 0$   then
a) $e\left(\lambda\right)$  decreases in the interval $\left[0,\infty\right)$
b) $\max e\left(\lambda\right)=\sqrt{1-\left(\frac{b}{a}\right)^{2}}$
c) $e\left(\lambda\right)$  has no minimum value
d) All of the Above

Explanation:

9. Tangents are drawn from point $\left(\frac{a^{2}}{\sqrt{a^{2}-b^{2}}},\sqrt{a^{2}+b^{2}}\right),$
to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$     with eccentricity e. Then
a) difference of slopes of two tangents is $\frac{2}{1/e-e}$
b) product of two slopes is $e^{2}$
c) sum of two slopes is independent of e
d) Both a and b

b) $\sqrt{3}/2$