Principal Axes of Ellipse

Q. A man running around a race course notes that the sum of the distances of two flag posts from him is always $8$ meters. The area of the path he encloses in square meters if the distance between flag posts is 4 meters is

1. $12\sqrt{3}\pi$ sq.units
2. $18\sqrt{3}\pi$ sq.units
3. $15\sqrt{3}\pi$ sq.units
4. $8\sqrt{3}\pi$ sq.units

Q. Number of points on the ellipse $\frac{x^2}{50}+\frac{y^2}{20}=1$ from which prependicular of tangents are drawn to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ is

Q. If $S$ and $S^{\prime }$ are two foci of the ellipse $16x^2+25y^2=400$ and $PSQ$ is a focal chord such that $SP=16,$ then $S^{\prime }Q=$

1. $\frac{16}{9}$
2. $\frac{74}{9}$
3. $\frac{37}{9}$
4. $\frac{32}{3}$

Q. If the curve $\frac{x^2}{4}+y^2=1$ and $\frac{x^2}{a^2}+y^2=1$ for suitable values of ${}^{\prime }{a}^{\prime }$ cut on four concyclic points, the equation of the circle passing through these points is

1. $x^2+y^2=2$
2. $x^2+y^2=1$
3. $x^2+y^2=4$
4. $x^2+y^2=3$

Q.

A rectangular garden has a perimeter of $120ft$.

How do you find an equation for the area of the rectangle as a function of the width, then determine the length and width of the rectangle which provide the maximum area?

Q. The equation of parabola whose latus rectum is $2$ units, axis is $x+y-2=0$ and tangent at the vertex is $x-y+4=0$ is given by

1. $(x-y-4{)}^2=4\sqrt{2}(x+y-2)$
2. $(x-y-4{)}^2=2\sqrt{2}(x-y+2{)}^2$
3. $(x+y-2{)}^2=2\sqrt{2}(x-y+4)$
4. $(x+y-2{)}^2=4\sqrt{2}(x-y+4{)}^2$

Q. Which among the following statements are CORRECT for an ellipse in the $xy$ plane whose focii are at $(9,20)$and $(49,55)$ and having $x$ axis as a tangent to it

1. Length of major axis is $85$ unit
2. Length of minor axis is $20\sqrt{11}$ unit
3. Co-ordiante of the point where $x$ axis touches the ellipse are $(97,0)$
4. $(4,8)$ lies inside the ellipse

Q. The slope of the tangent to the curve $y=f\left(x\right)$ at $(x,f\left(x\right))$ is $2x+1$.If the curve passes through the point $(1,2)$ then the area of the region bounded by the curve, the $x-$axis and the line $x=1$ is

1. $\frac{5}{6}$.sq.unit
2. $\frac{6}{5}$.sq.unit
3. $\frac{1}{6}$.sq.unit
4. $6$.sq.unit

Q. If $S$ and $S^{\prime }$ are two foci of the ellipse $16x^2+25y^2=400$ and $PSQ$ is a focal chord such that $SP=16,$ then $S^{\prime }Q=$

1. $\frac{16}{9}$
2. $\frac{74}{9}$
3. $\frac{37}{9}$
4. $\frac{32}{3}$

Q. Find the points on the curve $y=x^3$ at which the slope of the tangent is equal to the $y$ coordinate of the point.

Q. The area of the quadrilateral formed by the tangent from the point $(4,5)$ to the circle $x^2+y^2-4x-2y-c=0$ with a pair of radii joining the points of contacts of these tangents is $8$ sq. units. The value of $c$ is

1. $12$
2. $3$
3. $11$
4. $-1$

Q. Assertion(A): The tangent to the curve $y=x^3-x^2-x+2$ at (1, 1) is parallel to the x axis.
Reason(R): The slope of the tangent to the above curve at (1, 1) is zero.

1. Both A and R are true but R is not correct explanation of A
2. Both A and R are true R is the correct explanation of A
3. A is true but R is false
4. A is false but R is true

Q. The length of the major axis and the minor axis of the ellipse $2x^2+3y^2-4x-12y+13=0$ are and respectively.

1. $\sqrt{2}$
2. $\sqrt{3}$
3. $\frac{2}{\sqrt{3}}$
4. $\frac{3}{\sqrt{2}}$

Q.

Find the area under given curves and given lines.

(a) $y=x^2;x=1,x=2$ and X - axis (b) $y=x^4;x=1,x=5$ and X - axis.

Q. If $3x^2+2\lambda xy+3y^2+(6-\lambda )x+(2\lambda -6)y-21=0$ is the equation of circle then its radius is

1. $1$
2. $3$
3. $2\sqrt{2}$
4. $4$

Q. If the curve $\frac{x^2}{4}+y^2=1$ and $\frac{x^2}{a^2}+y^2=1$ for suitable values of ${}^{\prime }{a}^{\prime }$ cut on four concyclic points, the equation of the circle passing through these points is

1. $x^2+y^2=2$
2. $x^2+y^2=1$
3. $x^2+y^2=4$
4. $x^2+y^2=3$