If the normals of the parabola y 2 = 4 x drawn at the end points of its latus rectum are tangents to the circle ( x - 3 ) 2 + ( y + 2 ) 2 = r 2 , then the value of r 2 is [2015]

Question

If the normals of the parabola $y^{2} = 4 x$ drawn at the end points of its latus rectum are tangents to the circle ${(x - 3)}^{2} + {(y + 2)}^{2} = r^{2}$ , then the value of $r^{2}$ is [2015]

Smart Achievers · Accepted Answer

(2)

End points of latus rectum of $y^{2} = 4 x$ are $(1, \pm 2)$

Equation of normal to $y^{2} = 4 x$ at (1, 2) is

$y - 2 = - 1 (x - 1) \Rightarrow x + y - 3 = 0$

As it is tangent to circle ${(x - 3)}^{2} + {(y + 2)}^{2} = r^{2}$

$∴$ $| \frac{3 + (- 2) - 3}{\sqrt{2}} |$ $= r \Rightarrow r^{2} = 2$

Solution

Q.
If the normals of the parabola $y^{2} = 4 x$ drawn at the end points of its latus rectum are tangents to the circle ${(x - 3)}^{2} + {(y + 2)}^{2} = r^{2}$ , then the value of $r^{2}$ is [2015]

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Solution

Q. If the normals of the parabola y2=4x drawn at the end points of its latus rectum are tangents to the circle (x-3)2+(y+2)2=r2, then the value of r2 is [2015]

Ans. (2) End points of latus rectum of y2=4x are (1,±2) Equation of normal to y2=4x at (1, 2) is y-2=-1(x-1)⇒x+y-3=0 As it is tangent to circle (x-3)2+(y+2)2=r2 ∴ |3+(-2)-32|=r⇒r2=2

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Q.
If the normals of the parabola $y^{2} = 4 x$ drawn at the end points of its latus rectum are tangents to the circle ${(x - 3)}^{2} + {(y + 2)}^{2} = r^{2}$ , then the value of $r^{2}$ is [2015]