Point Form of Normal: Ellipse

Q. On the ellipse $\frac{x^2}{8}+\frac{y^2}{4}=1,$ let $P$ be a point in the second quadrant such that the tangent at $P$ to the ellipse is perpendicular to the line $x+2y=0.$ Let $S$ and $S^{\prime }$ be the foci of the ellipse and $e$ be its eccentricity. If $A$ is the area of the triangle $SP{S}^{\prime },$ then the value of $(5-e^2)\cdot A$ is

1. $12$
2. $14$
3. $6$
4. $24$

Q. The equation $\frac{x^2}{2-r}+\frac{y^2}{r-5}+1=0$ represent an ellipse if

1. 
2. 
3. 
4. 

Q. The locus of mid-points of the line segments joining $(-3,-5)$ and the points on the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ is

1. $36x^2+16y^2+90x+56y+145=0$
2. $9x^2+4y^2+18x+8y+145=0$
3. $36x^2+16y^2+72x+32y+145=0$
4. $36x^2+16y^2+108x+80y+145=0$

Q.

On the portion of the straight line, $\text{x+2y = 4}$intercepted between the axes, a square is constructed on the side of the line away from the origin. Then the point of intersection of its diagonals has co-ordinates

1. $2,3$

2. $3,2$

3. $3,3$

4. $2,2$

Q. The equation of the normal to the ellipse $\frac{x^2}{18}+\frac{y^2}{8}=1$ at the point (3, 2) is .

1. 3x - 2y = 5
2. 3x + 2y = 5
3. 2x - 3y = 5
4. 2x + 3y = 5

Q.

Find Minimize Z=3x+9y subject to

x+3y≤60 , x≤y and x, y≥0

Q. If the normal at an end point of a latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ passes through one extremity of the minor axis. Then the eccentricity of the ellipse is equal to

1. $\frac{\sqrt{5}-1}{2}$
2. $\frac{1}{5^{\frac{1}{4}}}$
3. $\frac{1}{2}$
4. $\sqrt{\frac{\sqrt{5}-1}{2}}$

Q. Show that the circle on the chord xcosα+ ysinα= p
Of the circle x^2+ y^2= a^2 as diameter is x^2+ y^2 -a^2-2p( xcosα+ ysinα- p)=0

Q. If normal at point $(6,2)$ to the ellipse passes through its nearest focus $(5,2),$ having centre at $(4,2),$ then its eccentricity is

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

Q. The polar of point (2t, t-4)w.r.t the circle, x*2+y*2-4x-6y+1=0 passes through the point

Q. The equation of normal at point $P(8\sqrt{2},1)$ on the ellipse $\frac{x^2}{144}+\frac{y^2}{9}=1$ is

1. $\sqrt{2}x+y=15$
2. $\sqrt{2}x-y=15$
3. $x+\sqrt{2}y=15$
4. $x-\sqrt{2}y=15$

Q. If the normal at the end of latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ of eccentricity e passes through one end of the minor axis then $e^4+e^2=$

1. 
2. 
3. 
4. 

Q.

Which could be the initial point of vector $\text{u}$ if it has a magnitude of $17$ and the terminal point $\left(3,13\right)$$?$

1. $(-5,-2)$

2. $(-5,2)$

3. $(5,-2)$

4. $(5,2)$

Q. The area enclosed by the curve $y=9-x^2$ and the positive coordinate axes is

Q. If the area bounded by the curve $y=2^{kx},k>0$ and $x=0$, $x=2$ and $x-$ axis is $\frac{3}{\ln 2},$ then the value of $k$ is

Q. If the obtuse angle bisector and angular bisector contains origin is same for the pair of lines $2x+3y-\lambda =0$ and $3x+2y-(\lambda +2)=0$ then $\lambda$ belongs to

1. $(-2,0)$
2. $(-\infty ,-2)\cup (0,\infty )$
3. $(-\infty ,-2]\cup [0,\infty )$
4. None of the above

Q.

In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

(a)z = 2 (b)

(c) (d)5y + 8 = 0

Q. The equation of normal at point $P(8\sqrt{2},1)$ on the ellipse $\frac{x^2}{144}+\frac{y^2}{9}=1$ is

1. $\sqrt{2}x+y=15$
2. $\sqrt{2}x-y=15$
3. $x+\sqrt{2}y=15$
4. $x-\sqrt{2}y=15$

Q. The area bounded by the curve $y=\frac{1}{x^2}$ and its asymptote between the lines $x=1$ to $x=3$ is

1. $\frac{1}{3}$
2. $\frac{2}{3}$
3. $\frac{1}{2}$
4. $\frac{1}{6}$

Q.
​​​​​​$\begin{array}{cc}ColumnI& ColumnII\\ \text{a. If}x,y\in R,\text{satisfying the equation}\frac{(x-4{)}^2}{4}+\frac{y^2}{9}=1& \text{p.}-\frac{2}{3}\\ \text{then the difference between the largest and smallest value}& \\ \text{of the expression}\frac{x^2}{4}+\frac{y^2}{9}\text{is}& \\ \text{b. If}PQ\text{is focal chord of ellipse}\frac{x^2}{25}+\frac{y^2}{16}=1\text{which passes}& \text{q.}10\\ \text{through}S\equiv (3,0)\text{and}PS=2,\text{then length of chord}PQ\text{is}& \\ \text{c. If the normal at the point}P\left(\theta \right)\text{to the ellipse}\frac{x^2}{14}+\frac{y^2}{5}=1& \\ \text{intersect it again at the point}Q\left(2\theta \right),\text{then the value of}& \\ \cos \theta \text{is}& \text{r.}\frac{3}{4}\sqrt{7}\\ \text{d. The length of common tangent to}{x}^2+y^2=16\text{and}& \text{s.}8\\ 9x^2+25y^2=225\text{is}& \end{array}$

1. $a-s;b-r;c-q;d-p$
2. $a-p;b-q;c-r;d-s$
3. $a-s;b-q;c-p;d-r$
4. $a-q;b-r;c-s;d-p$

Q. 41. Let the line x-2/3=y-1/-5=z+2/2 lie in the plane x+3y-az+b=0 then (a, b) is

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

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

Q. The eccentricity of ellipse $4x^2+9y^2=36$ is

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

Q. y=x2^1/2- 4a2^1/2 is a normal chord to y² = 4ax. Then it’s length is??

Q. The locus of the point of intersection of the tangents to the circle $x^2+y^2=a^2$ at the points whose parametric angle differ by $\frac{\pi }{3}$ is

1. $2(x^2+y^2)=4a^2$
2. $2(x^2+y^2)=a^2$
3. $3(x^2+y^2)=4a^2$
4. $3(x^2+y^2)=a^2$

Q. ​​​​​​$\begin{array}{cc}ColumnI& ColumnII\\ \text{a. If}x,y\in R,\text{satisfying the equation}\frac{(x-4{)}^2}{4}+\frac{y^2}{9}=1& \text{p.}-\frac{2}{3}\\ \text{then the difference between the largest and smallest value}& \\ \text{of the expression}\frac{x^2}{4}+\frac{y^2}{9}\text{is}& \\ \text{b. If}PQ\text{is focal chord of ellipse}\frac{x^2}{25}+\frac{y^2}{16}=1\text{which passes}& \text{q.}10\\ \text{through}S\equiv (3,0)\text{and}PS=2,\text{then length of chord}PQ\text{is}& \\ \text{c. If the normal at the point}P\left(\theta \right)\text{to the ellipse}\frac{x^2}{14}+\frac{y^2}{5}=1& \\ \text{intersect it again at the point}Q\left(2\theta \right),\text{then the value of}& \\ \cos \theta \text{is}& \text{r.}\frac{3}{4}\sqrt{7}\\ \text{d. The length of common tangent to}{x}^2+y^2=16\text{and}& \text{s.}8\\ 9x^2+25y^2=225\text{is}& \end{array}$

1. $a-s;b-r;c-q;d-p$
2. $a-p;b-q;c-r;d-s$
3. $a-s;b-q;c-p;d-r$
4. $a-q;b-r;c-s;d-p$

Q. If the normal at the end of latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ of eccentricity e passes through one end of the minor axis then $e^4+e^2=$

1. $0$
2. $1$
3. $-1$
4. $2$

Q. A chord $MP$ parallel to the latus rectum of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ with centre at $O(0,0)$ intersects the auxiliary circle at $Q$. Then the locus of the point of intersection of normals at $P$ and $Q$ to the respective curve is

1. $x^2+y^2=8$
2. $x^2+y^2=34$
3. $x^2+y^2=64$
4. $x^2+y^2=15$

Q. A chord $MP$ parallel to the latus rectum of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ with centre at $O(0,0)$ intersects the auxiliary circle at $Q$. Then the locus of the point of intersection of normals at $P$ and $Q$ to the respective curve is

1. $x^2+y^2=8$
2. $x^2+y^2=34$
3. $x^2+y^2=64$
4. $x^2+y^2=15$

Q. A chord $MP$ parallel to the latus rectum of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ with centre at $O(0,0)$ intersects the auxiliary circle at $Q$. Then the locus of the point of intersection of normals at $P$ and $Q$ to the respective curve is

1. $x^2+y^2=8$
2. $x^2+y^2=34$
3. $x^2+y^2=64$
4. $x^2+y^2=15$