A triangle is formed by the tangents at the point $(2,2)$ on the curves $y^2=2 x$ and $x^2+y^2=4 x$, and the line $x+y+2=0$. If $r$ is the radius of its circumcircle, then $r ^2$ is equal to $........$.
A triangle is formed by the tangents at the point $(2,2)$ on the curves $y^2=2 x$ and $x^2+y^2=4 x$, and the line $x+y+2=0$. If $r$ is the radius of its circumcircle, then $r ^2$ is equal to $........$.
- [JEE MAIN 2023]
- A
$10$
- B
$18$
- C
$15$
- D
$14$
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Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E: \frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P: y^2=12 x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the point $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?
($A$) The area of the quadrilateral $A_1 A _2 A _3 A _4$ is $35$ square units
($B$) The area of the quadrilateral $A_1 A_2 A_3 A_4$ is $36$ square units
($C$) The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-3,0)$
($D$) The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-6,0)$
Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E: \frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P: y^2=12 x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the point $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?
($A$) The area of the quadrilateral $A_1 A _2 A _3 A _4$ is $35$ square units
($B$) The area of the quadrilateral $A_1 A_2 A_3 A_4$ is $36$ square units
($C$) The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-3,0)$
($D$) The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-6,0)$
- [IIT 2023]
Let $L$ be a tangent line to the parabola $y^{2}=4 x-20$ at $(6,2)$ . If $L$ is also a tangent to the ellipse $\frac{ x ^{2}}{2}+\frac{ y ^{2}}{ b }=1,$ then the value of $b$ is equal to ..... .
Let $L$ be a tangent line to the parabola $y^{2}=4 x-20$ at $(6,2)$ . If $L$ is also a tangent to the ellipse $\frac{ x ^{2}}{2}+\frac{ y ^{2}}{ b }=1,$ then the value of $b$ is equal to ..... .
- [JEE MAIN 2021]
Let a line $L$ pass through the point of intersection of the lines $b x+10 y-8=0$ and $2 x-3 y=0$, $b \in R -\left\{\frac{4}{3}\right\}$. If the line $L$ also passes through the point $(1,1)$ and touches the circle $17\left( x ^{2}+ y ^{2}\right)=16$, then the eccentricity of the ellipse $\frac{x^{2}}{5}+\frac{y^{2}}{b^{2}}=1$ is.
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- [JEE MAIN 2022]