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Q.

The focus of the parabola is (1,1) and the tangent at the vertex has the equation x+y=1 which of the following is correct

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a

Equation of parabola is xy2=4x+y1

b

The length of the latursectum is 22

c

The coordinates of the vertex 12,12

d

Equation of parabola is xy2=2x+y1

answer is B, C, D.

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Detailed Solution

The distance from focus to the tangent at vertex  is a=2 

Directrix is a line parallel to the tangent at vertex and at a distance of 22 units from focus

Directrix is passing through the image of focus in tangent at vertex

Hence the equation of the directrix is x+y=0

By using the definition of the parabola, the locus is 

x+y22=x12+y12

x2+y2+2xy=2x2+2y24x4y+4=0

xy2=4x+y1

This is the equation of the parabola

Vertex is the foot of the perpendicular of focus on the tangent at vertex

here vertex is 12,12

a=12, hence the length of the latusrectum is 22

 

 

 

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