Question

# Let S be a curved mirror passing through (3,4) having the property that all the light emerging from origin(focus) , after getting reflected from the mirror becomes parallel to x−axis. The angle between the tangents drawn from the point (−2,6) is 90∘. If the circle (x−4)2+y2=r2 internally touches the curve S, then the value of r2 is

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Solution

## In any parabola , if we send a ray from the focus to the mirror curve ,it will be parallel to axis of the parabola after reflection. So, (0,0) is the focus ,X−axis is the axis The equation of the parabola be y2=4a(x−b) It passes through (3,4). ⇒16=4a(3−b)⇒a=43−b The locus of the point from which perpendicular tangents are drawn to the parabola is directrix. Directrix is perpendicular to the axis and passes through (−2,6). So, the equation of the directrix is x=−2 Vertex is the midpoint of A(−2,0) and S(0,0) ⇒b=−1⇒a=1 The equation of the parabola is y2=4(x+1) Solving parabola and circle (x−4)2+4(x+1)=r2 ⇒x2−4x+20−r2=0 If both curve touch each other ,then △=0 16−4(20−r2)=0 ⇒20−r2=4⇒r2=16

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