The equation of the common tangent to the cir | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) For the parabola $y^2 = 8x$,we have $4a = 8$,so $a = 2$. Any tangent to the parabola is given by $y = mx + \frac{a}{m}$,which simplifies to $mx -

Vedclass · Accepted Answer

$y = x + 2$

Vedclass · Answer

(B) For the parabola $y^2 = 8x$,we have $4a = 8$,so $a = 2$. Any tangent to the parabola is given by $y = mx + \frac{a}{m}$,which simplifies to $mx - y + \frac{2}{m} = 0$. If this line is a tangent to the circle $x^2 + y^2 = 2$,the perpendicular distance from the center $(0, 0)$ to the line must equal the radius $r = \sqrt{2}$. Thus,$\frac{|2/m|}{\sqrt{m^2 + 1}} = \sqrt{2}$. Squaring both sides,we get $\frac{4}{m^2} = 2(m^2 + 1)$,which leads to $2 = m^2(m^2 + 1)$. $m^4 + m^2 - 2 = 0$. Factoring gives $(m^2 + 2)(m^2 - 1) = 0$. Since $m^2$ must be positive,$m^2 = 1$,so $m = \pm 1$. Substituting $m = 1$ into the tangent equation,we get $y = x + \frac{2}{1} = x + 2$. Substituting $m = -1$,we get $y = -x - 2$.

The equation of the common tangent to the circle $x^2 + y^2 = 2$ and the parabola $y^2 = 8x$ is

Similar Questions

Which of the following oxides reacts with both $HCl$ and $NaOH$?

The coefficient of $x^3 y^4 z^5$ in the expansion of $(x y+y z+x z)^6$ is

What is the similarity between $DNA$ and $RNA$?

Which series possesses an inferior ovary?

In the circuit shown in the figure,the current $I$ is: (in $\,A$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module