Position of a Point W.R.T Ellipse

Q. If the tangent at the point $P\left(\theta \right)$ to the ellipse $16x^2+11y^2=256$ is also a tangent to the circle $x^2+y^2-2x=15,$ then possible value(s) of $\theta$ is/are

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

Q. A tangent to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ at any point $P$ meets the line $x=0$ at a point $Q.$ Let $R$ be the image of $Q$ in the line $y=x$, then the circle whose extremities of a diameter are $Q$ and $R$ passes through a fixed point. The fixed point is

1. $(3,0)$
2. $(5,0)$
3. $(0,0)$
4. $(4,0)$

Q. The line $3x+4y=\sqrt{7}$ touches the ellipse $3x^2+4y^2=1$ at the point

1. $(\sqrt{7},\sqrt{7})$
2. $(\frac{1}{\sqrt{7}},-\frac{1}{\sqrt{7}})$
3. $(\frac{1}{\sqrt{7}},\frac{1}{\sqrt{7}})$
4. $(-\sqrt{7},\sqrt{7})$

Q. The slope of common tangent to the curves $x^2+y^2=16$ and $\frac{x^2}{25}+\frac{y^2}{4}=1$ in the $1^{\text{st}}$ quadrant is $m$, then $3m^2=$

Q. 20. A tangent to the curve, y = f(x) at P(x, y) meets x-axis at A and y-axis at B. If AP : BP = 1 : 3 and f(1)= 1, then the curve also passes through the point : A) (1/3 , 23) B) (3 , 1/28) C) (1/2 , 3) D) (1, 2/8)

Q.

Latus rectum of the parabola whose focus is (3, 4) and whose tangent at vertex has the equation x + y = 7 + 5√2 is ?

Q. If tangents drawn to the ellipse at the parametric point $\theta$, where $\tan \theta =2$ meets the auxillary circle at $P$ and $Q$ and $PQ$ subtends rightangle at the centre of the ellipse, then eccentricity of the ellipse is

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

Q.

Locus of the point of intersection of perpendicular tangents to the circle $x^2+y^2=16$ is

1. $x^2+y^2=8$

2. $x^2+y^2=32$

3. $x^2+y^2=64$

4. $x^2+y^2=16$

Q. An ellipse with major and minor axis length as $2a$ and $2b$ units touches coordinate axis in first quadrant. If foci are $(x_1,y_1)$ and $(x_2,y_2),$ then the value of $x_1{x}_2+y_1{y}_2$ is

1. $2b^2$
2. $a^2{b}^2$
3. $a^2+b^2$
4. $2a^2$

Q. If the point $P(\alpha ,-\alpha )$ lies inside the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1,$ then

1. $\alpha \in (-\infty ,-\frac{12}{5})\cup (\frac{12}{5},\infty )$
2. $\alpha \in (-\frac{12}{5},\frac{12}{5})$
3. $\alpha \in (-\frac{5}{12},\frac{5}{12})$
4. $\alpha \in (-\infty ,-\frac{5}{12})\cup (\frac{5}{12},\infty )$

Q. Equation of the tangent to y²= 6x at the positive end of the latusrectum is.?

Q. The position of the point $(2,-3)$ with respect to the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1,$ is

1. It lies on the ellipse
2. It lies outside the ellipse
3. It lies inside the ellipse
4. None of these

Q.

find the area of the triangle formed by the positive x axis and the tangent and normal to the curve x^2 + y^2 =9 at(2, √5).

Q. Consider the curves $C_1:\frac{x^2}{9}+\frac{y^2}{4}=1,$ $C_2:(x+4{)}^2+y^2=1$ and $C_3:y^2=4a(x-3)$. If all three curves have two common tangents, then

1. Length of latus rectum of curve $C_3$ is $\frac{24}{7}$ units.
2. $x-$intercept of common tangents is $-\frac{23}{3}$.
3. Area of the triangle formed by the common tangents and $y-$axis is $\frac{529}{12\sqrt{7}}$ sq. units.
4. Product of the length of the perpendiculars drawn from $(-\sqrt{5},0)$ and $(\sqrt{5},0)$ to any of the common tangents is equal to $4$.

Q. If $y=f\left(x\right)$ and $y=g\left(x\right)$ are symmetrical about the line $x=\frac{\alpha +\beta }{2},$ then $\underset{\alpha }{\overset{\beta }{\int }}f\left(x\right)g^{\prime }\left(x\right)dx$ is equal to

1. $\underset{\alpha }{\overset{\beta }{\int }}{f}^{\prime }\left(x\right)g\left(x\right)dx$
2. $-\underset{\alpha }{\overset{\beta }{\int }}{f}^{\prime }\left(x\right)g\left(x\right)dx$
3. $\frac{1}{2}\underset{\alpha }{\overset{\beta }{\int }}\left(f\right(x\left)g^{\prime }\right(x)-f^{\prime }(x\left)g\right(x\left)\right)dx$
4. $\frac{1}{2}\underset{\alpha }{\overset{\beta }{\int }}\left(f\right(x\left)g^{\prime }\right(x)+f^{\prime }(x\left)g\right(x\left)\right)dx$

Q. If a tangent of slope 2 of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is normal to the circle $x^2+y^2+4x+1=0$ then the maximum value of ab is

1. 2
2. 4
3. 6
4. cannot be found

Q. A circle of radius $r(<a)$ is concentric with ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$, then slope of the common tangents to ellipse and circle is

1. $\pm \sqrt{\frac{r^2+b^2}{a^2+r^2}}$
2. $\pm \sqrt{\frac{r^2+b^2}{a^2-r^2}}$
3. $\pm \sqrt{\frac{r^2-b^2}{a^2+r^2}}$
4. $\pm \sqrt{\frac{r^2-b^2}{a^2-r^2}}$

Q.

If for the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,S_1=\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}-1$, for point $(x_1,y_1)$, Which of the following is true

1. $S_1$$>0$$\Rightarrow (x_1,y_1)$ is inside the ellipse
2. $S_1=0\Rightarrow (x_1,y_1)$ is on the ellipse
3. $S_1$$<0$$\Rightarrow (x_1,y_1)$is outside the ellipse
4. $S_1$$>0$$\Rightarrow (x_1,y_1)$ is outside the ellipse

Q. If the lines (y-b)=$m_1$ (x+a) and (y-b)= $m_2$ (x+a) are the tangents of $y^2=4ax$then

1. 
2. 
3. 
4. 

Q. $\begin{array}{cccc}& ListI& & ListII\\ P.& \text{Let}y\left(x\right)=\cos \left(3{\cos }^{-1}x\right),x\in [-1,1],x\ne \pm \frac{\sqrt{3}}{2}.\text{Then}\frac{1}{y\left(x\right)}\left\{\right(x^2-1)\frac{d^{2}y\left(x\right)}{d{x}^2}+x\frac{dy\left(x\right)}{dx}\}\text{equals}& 1.& 1\\ Q.& \text{Let}{A}_1,A_2,\cdots ,A_n(n>2)\text{be the vertices of a regular polygon of}n\text{sides with its centre at the origin. Let}\overrightarrow{a_k}\text{be the position vector of the point}{A}_k,k=1,2,\cdots ,n.\text{If}\left|\sum _{k=1}^{n-1}(\overrightarrow{a_k}\times \overrightarrow{a_{k+1}})\right|=\left|\sum _{k=1}^{n-1}(\overrightarrow{a_k}\cdot \overrightarrow{a_{k+1}})\right|,\text{then}\text{the minimum value of}n\text{is}& 2.& 2\\ R.& \text{If the normal from the point}P(h,1)\text{on the ellipse}\frac{x^2}{6}+\frac{y^2}{3}=1\text{is perpendicular to the line}x+y=8,\text{then the value of}h\text{is}& 3.& 8\\ S.& \text{Number of positive solutions satisfying the equation}{\tan }^{-1}\left(\frac{1}{2x+1}\right)+{\tan }^{-1}\left(\frac{1}{4x+1}\right)={\tan }^{-1}\left(\frac{2}{x^2}\right)\text{is}& 4.& 9\end{array}$

Which of the following option is correct?

1. $\left(P\right)\to \left(4\right),\left(Q\right)\to \left(3\right)\left(R\right)\to \left(2\right),\left(S\right)\to \left(1\right)$
2. $\left(P\right)\to \left(2\right),\left(Q\right)\to \left(4\right)\left(R\right)\to \left(3\right),\left(S\right)\to \left(1\right)$
3. $\left(P\right)\to \left(4\right),\left(Q\right)\to \left(3\right)\left(R\right)\to \left(1\right),\left(S\right)\to \left(2\right)$
4. $\left(P\right)\to \left(2\right),\left(Q\right)\to \left(4\right)\left(R\right)\to \left(1\right),\left(S\right)\to \left(3\right)$

Q. The point (4, -3) lies inside the ellipse $5x^2+7y^2=140$.

1. False
2. True

Q. The number of tangents that can be drawn to the ellipse $16x^2+9y^2=144$ from the point (3, -4) is .

1. 2
2. 3
3. \N
4. 1

Q. An ellipse with major and minor axis length as $2a$ and $2b$ units touches coordinate axis in first quadrant. If foci are $(x_1,y_1)$ and $(x_2,y_2),$ then the value of $x_1{x}_2+y_1{y}_2$ is

1. $2a^2$
2. $2b^2$
3. $a^2{b}^2$
4. $a^2+b^2$

Q. $$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(A)}& \text{If real numbers}x\text{and}y\text{satisfy}& & \\ & (x+5{)}^2+(y-12{)}^2=81,& & \\ & \text{then the minimum value of}\sqrt{x^2+y^2}\text{is}& \text{(P)}& 1\\ \text{(B)}& \text{The line}3x+6y=k\text{intersects the curve}& & \\ & 2x^2+2xy+3y^2=1\text{at point}A\text{and}B.& & \\ & \text{If the circle with}AB\text{as a diameter passes}& & \\ & \text{through the origin, then the value of}{k}^2\text{is}& \text{(Q)}& 2\\ \text{(C)}& \text{If two perpendicular tangents can be}& & \\ & \text{drawn from the origin to the circle}& & \\ & x^2-6x+y^2-2py+17=0\text{, then}& & \\ & \text{the value of}|p|\text{is}& \text{(R)}& 4\\ \text{(D)}& \text{If the circles}& & \\ & x^2+y^2+(3+\sin \beta )x+\left(2\cos \alpha \right)y=0\text{and}& & \\ & x^2+y^2+\left(2\cos \alpha \right)x+2cy=0\text{touch each}& & \\ & \text{other, then the maximum value of}{}^{\prime }{c}^{\prime }\text{is}& \text{(S)}& 5\\ & & \text{(T)}& 7\\ & & \text{(U)}& 9\end{array}$$
Which of the following is the only CORRECT combination?

1. $\left(C\right)\to \left(R\right),\left(D\right)\to \left(Q\right)$
2. $\left(C\right)\to \left(S\right),\left(D\right)\to \left(P\right)$
3. $\left(C\right)\to \left(P\right),\left(D\right)\to \left(S\right)$
4. $\left(C\right)\to \left(T\right),\left(D\right)\to \left(P\right)$

Q.

The line 2x-y+1=0 is tangent to circle at (2, 5) and the Venter of the circle lies on x-2y=4.then find radius of circle

Q. S (3, 4)and s(9, 12)are the fici of ellipse and foot of perpendicular from s to tangent on ellipse is (1, -4) then eccentricity of ellipse is

Q. The slope of common tangent to the curves $x^2+y^2=16$ and $\frac{x^2}{25}+\frac{y^2}{4}=1$ in the $1^{\text{st}}$ quadrant is $m$, then $3m^2=$

Q. $$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(A)}& \text{If real numbers}x\text{and}y\text{satisfy}& & \\ & (x+5{)}^2+(y-12{)}^2=81,& & \\ & \text{then the minimum value of}\sqrt{x^2+y^2}\text{is}& \text{(P)}& 1\\ \text{(B)}& \text{The line}3x+6y=k\text{intersects the curve}& & \\ & 2x^2+2xy+3y^2=1\text{at point}A\text{and}B.& & \\ & \text{If the circle with}AB\text{as a diameter passes}& & \\ & \text{through the origin, then the value of}{k}^2\text{is}& \text{(Q)}& 2\\ \text{(C)}& \text{If two perpendicular tangents can be}& & \\ & \text{drawn from the origin to the circle}& & \\ & x^2-6x+y^2-2py+17=0\text{, then}& & \\ & \text{the value of}|p|\text{is}& \text{(R)}& 4\\ \text{(D)}& \text{If the circles}& & \\ & x^2+y^2+(3+\sin \beta )x+\left(2\cos \alpha \right)y=0\text{and}& & \\ & x^2+y^2+\left(2\cos \alpha \right)x+2cy=0\text{touch each}& & \\ & \text{other, then the maximum value of}{}^{\prime }{c}^{\prime }\text{is}& \text{(S)}& 5\\ & & \text{(T)}& 7\\ & & \text{(U)}& 9\end{array}$$
Which of the following is the only CORRECT combination?

1. $\left(A\right)\to \left(R\right),\left(B\right)\to \left(U\right)$
2. $\left(A\right)\to \left(S\right),\left(B\right)\to \left(P\right)$
3. $\left(A\right)\to \left(Q\right),\left(B\right)\to \left(U\right)$
4. $\left(A\right)\to \left(R\right),\left(B\right)\to \left(Q\right)$

Q.

Find the set of values of $\alpha$ for which point the $P(\alpha ,-\alpha )$ is inside
$\frac{x^2}{16}+\frac{y^2}{9}=1$

1. $(\frac{-12}{5},\frac{12}{5})$
2. $(\frac{-13}{5},\frac{13}{5})$
3. $[\frac{-12}{5},\frac{12}{5}]$
4. $(\frac{-13}{5},\frac{13}{5}]$

Q.

If distance between the directrices be thrice the distance between the foci, then the eccentricity of ellipse is

1. 1/2

2. 2/3

3. 

4. 4/5