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04%20-%20Parabola

# 04%20-%20Parabola - PROBLEMS(1 04 PARABOLA Page 1 Find the...

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PROBLEMS 04 - PARABOLA Page 1 ( 1 ) Find the co-ordinates of the focus, length of the latus-rectum and equation of the directrix of the parabola x 2 = - 8y. [ Ans: ( 0, - 2 ), 8, y = 2 ] ( 2 ) If the line 3x + 4y + k = 0 is a tangent to the parabola y 2 = 12x, then find k and obtain the co-ordinates of the point of contact. = 8 , 3 16 16, k : Ans - ( 3 ) Derive the equations of the tangents drawn from the point ( 1, 3 ) to the parabola y 2 = 8x. Obtain the co-ordinates of the point of contact. + = + = 2 , 2 1 at 1 2x y and ) 4 2, ( at 2 x y : Ans ( 4 ) Find the equation of the chord of the parabola joining the points P ( t 1 ) and Q ( t 2 ). If this chord passes through the focus, then prove that t 1 t 2 = - 1. [ Ans: ( t 1 + t 2 )y = 2 ( x + a t 1 t 2 ) ] ( 5 ) If one end-point of a focal chord of the parabola y 2 = 16x is ( 9, 12 ), then find its other end-point. 3 16 , 9 16 : Ans - ( 6 ) The points P ( t 1 ), Q ( t 2 ) and R ( t 3 ) are on the parabola y 2 = 4ax. Show that the area of triangle PQR is a 2 l ( t 1 - t 2 ) ( t 2 - t 3 ) ( t 3 - t 1 ) l . ( 7 ) If the focus of the parabola y 2 = 4ax divides a focal chord in the ratio 1 : 2, then find the equation of the line containing this focal chord. [ Ans: y = ± 2 2 ( x - a ) ]

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PROBLEMS 04 - PARABOLA Page 2 ( 8 ) If a focal chord of the parabola y 2 = 4ax forms an angle of measure θ with the positive X-axis, then show that its length is 4 l a l cosec 2 θ . ( 9 ) Show that the length of the focal chord of the parabola y 2 = 4ax at the point P ( t ) is l a l 2 t 1 t + ( 10 ) Find the condition for the line x cos α + y sin α = p to be a tangent to the
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