Interaction between Ellipse and a Line

If a line $px+qy=12$ is a tangent to the ellipse $4x^2+9y^2=16$, then $pq$ can not be equal to :

If the line $y=2x+c$ touches the ellipse $\frac{x^2}{8}+\frac{y^2}{4}=1$, then $c=$

The number of possible tangents which can be drawn to the curve $4x^2-9y^2=36$, which are perpendicular to the straight line $5x+2y-10=0$ is

On the ellipse $\frac{{\mathrm{x}}^2}{18}+\frac{{\mathrm{y}}^2}{8}=1,$ then Find the point $\mathrm{M}$nearest to the line $2 \mathrm{x}-3 \mathrm{y}+25=0$.

A line perpendicular to the join of the points $\left(\frac{11}{3}, 0\right)$ and $\left(0, \frac{11}{4}\right)$, and which is tangent to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ is ____.

Cotyledons are also called-

An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4.$ One vertex of the triangle is $\left(\mathrm{0,1}\right)$ and one altitude is contained in the $y$-axis. If the length of each side is $k\sqrt{3}$ units, then $k$ is

The tangents at the extremities of the latus rectum of the ellipse $3x^2+4y^2=12$ form a rhombus $\mathrm{P}\mathrm{Q}\mathrm{R}\mathrm{S}$. Area (in sq. units) of the rhombus $\mathrm{P}\mathrm{Q}\mathrm{R}\mathrm{S}$ outside the ellipse is equal to

The values that m can take so that the straight line $y=4x+m$ touches the curve $x^2+4y^2=4$ is

The line $x={at}^2$, meets the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in the real points, if

The line passing through 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

Equation of tangents to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, which are perpendicular to the line $3x+4y=7$, are

Equation of tangents to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, which are perpendicular to the line $3x+4y=7$, are

Let $AOB$ be the positive quadrant of ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where $OA=a \& OB=b$. Then the area enclosed by the line $bx+ay-ab=0$ and the ellipse is

Equation of tangents to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, which are perpendicular to the line $3x+4y=7$, are

Equation of tangents to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, which are perpendicular to the line $3x+4y=7$, are

The number of values of $c$ such that the straight line $y=4x+c$ touches the curve $\frac{x^2}{4}+y^2=1$ is