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Question

# Two tangents on a parabola are x−y=0 and x+y=0. Let (2,3) is the focus of the parabola, then

A
The equation of the tangent at vertex is 4x6y+5=0
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B
The equation of the tangent at vertex is 4x6y+1=0
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C
The length of the latus rectum of the parabola is 1052
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D
The length of the latus rectum of the parabola is 1013
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Solution

## The correct options are A The equation of the tangent at vertex is 4x−6y+5=0 D The length of the latus rectum of the parabola is 10√13 The foot of the perpendicular from the focus upon any tangent lies on the tangent at the vertex. The foot of perpendicular of (2,3) on the line x−y=0 is A(52,52). The foot of of perpendicular of (2,3) on the line x+y=0 is B(−12,12). The equation of tangent at vertex (Eq. of AB) is 4x−6y+5=0 Let the length of the latus rectum be 4a Then the length of perpendicular to the tangent at vertex from the focus is a. a=|4(2)−6(3)+5|√52⇒4a=10√13

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