## Ellipse Questions and Answers Part-2

1. A line of fixed length a+b moves so that its ends are always on two fixed perpendicular straight lines . then the locus of a point , which divided the line into two parts of length a and b is
a) a parabola
b) a circle
c) an ellipse
d) none of these

Explanation: Let coordinates of P that divide AB in the ratio a : b be (h, k)

2. Two tangents are drawn to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$    from a point P (h, k) . if the point where these tangents meet the axes are concyclic , then locus of P is
a) $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$
b) $x^{2}+y^{2}=a^{2}+b^{2}$
c) xy=ab
d) $x^{2}-y^{2}=a^{2}-b^{2}$

Explanation: Equation of tangents to the ellipse from point P(h,k) is

3. From a point P two perpendicular tangents PQ and PR are drawn to the ellipse $x^{2}+4y^{2}=4.$     Locus of circumcentre of $\triangle PQR$  is
a) $x^{2}+y^{2}=\frac{5}{4}\left(x^{2}+4y^{2}\right)$
b) $x^{2}+y^{2}=\frac{5}{16}\left(x^{2}+4y^{2}\right)^{2}$
c) $x^{2}+4y^{2}=16$
d) $x^{2}+4y^{2}=\left(x^{2}+y^{2}-4\right)^{2}$

Explanation:

4. Let AB be a chord of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,$    which subtend a right at the centre of the ellipse. If L is the foot of perpendicular from the origin to AB, then locus of L is
a) a circle with centre at the origin
b) an ellipse with length of major axis a + b
c) a circle with centre at (a, 0)
d) none of these

Explanation:

5. The line passing throught the extremity A of the major axis and extremity B of the minor axis of the ellipse $x^{2}+9y^{2}=9$    meets its auxiliary circle at the point M. Then the area of the triangle with vertices at A,M and the origin O is
a) $\frac{31}{10}$
b) $\frac{29}{10}$
c) $\frac{21}{10}$
d) $\frac{27}{10}$

Explanation: Equation of given ellipse is

6. The area of the quadrilateral formed by the tangents at the end points of the latus recta of the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$
is
a) $\frac{27}{4}$
b) 9
c) $\frac{27}{2}$
d) 27

Explanation:

7. Locus of the mid points of the segments which are tangents to the ellipse $\frac{x^{2}}{2}+\frac{y^{2}}{1}=1$
and which are intercepted between the coordinate axes is
a) $\frac{x^{2}}{2}+\frac{y^{2}}{4}=1$
b) $\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$
c) $\frac{1}{3x^{2}}+\frac{1}{4y^{2}}=1$
d) $\frac{1}{2x^{2}}+\frac{1}{4y^{2}}=1$

Explanation: Equation of given ellipse is

8. Consider the ellipse $x^{2}+2y^{2}=2$   . Let L be the end of the latus rectum in the first quadrant. The tangent at L to the ellipse meets the right side directrix at T. The normal at L meets the major axis at N. Area of the triangle LNT in sq. units is
a) $\frac{3}{4}$
b) $\frac{3\sqrt{2}}{4}$
c) $\frac{3}{4\sqrt{2}}$
d) $\frac{4}{3\sqrt{2}}$

Explanation:

9. 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 PQ, then the locus of M intersects the latus rectums of the given ellipse at points
a) $\left(\pm\frac{3\sqrt{5}}{2},\pm\frac{2}{7}\right)$
b) $\left(\pm\frac{3\sqrt{5}}{2},\pm\frac{\sqrt{19}}{4}\right)$
c) $\left(\pm2\sqrt{3},\pm\frac{1}{7}\right)$
d) $\left(\pm2\sqrt{3},\pm\frac{4\sqrt{3}}{7}\right)$

10. The ellipse $E_{1}:\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$    is inscribed in a rectangle R whose sides are parallel to the coordinate axes . Another ellipse $E_{2}$ passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse $E_{2}$ is
a) $\frac{\sqrt{2}}{2}$
b) $\frac{\sqrt{3}}{2}$
c) $\frac{1}{2}$
d) $\frac{3}{4}$