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Page 1 JEE Solved Example on Parabola JEE Mains Q1: The length of the chord intercepted by the parabola ?? ?? = ?? ?? on the straight line ?? + ?? = ?? is- (A) 4 (B) ?? v ?? (C) 8 (D) ?? v ?? Ans: (C) ?? = 1 - ?? ? ?? 2 = 4 - 4?? ? ?? 2 + 4?? - 4 = 0 ( ?? 2 - ?? 1 ) 2 = ( 4) 2 + 4 × 4 = 32 ? |?? 2 - ?? 1 | = 4v 2 |?? 2 - ?? 1 | = ( 1 - ?? 2 )- ( 1 - ?? 1 ) = |?? 2 - ?? 1 | ? Length of chord = v 2 × |?? 2 - ?? 1 | = 8 Q2: A parabola is drawn with its focus at ( ?? , ?? ) and vertex at the focus of the parabola ?? ?? - ???? ?? - ?? ?? + ?? = 0 . The equation of the parabola is - (A) ?? ?? - ?? ?? - ?? ?? + ???? = ?? (B) ?? ?? - ?? ?? - ?? ?? + ???? = ?? (C) ?? ?? - ?? ?? + ?? ?? - ???? = ?? (D) ?? ?? + ?? ?? - ?? ?? - ???? = ?? Ans: (A) ?? 2 - 12?? - 4?? + 4 = 0 ( ?? - 2) 2 = 12?? Let ?? = ?? and ?? = ?? + 2 ? ?? 2 = 12?? Focus of parabola is ( 3,0) ? Focus in original coordinate system is ( 3,2) ? Vertex of new parabola is ( 3,2) &focus = ( 3,4) ? ?? = 2 Let ?? = ?? + 3&?? = ?? + 2 ? Equation is ?? 2 = 8?? ? ( ?? - 3) 2 = 8( ?? - 2) ? ?? 2 - 6?? - 8?? + 25 = 0 is the equation of parabola Page 2 JEE Solved Example on Parabola JEE Mains Q1: The length of the chord intercepted by the parabola ?? ?? = ?? ?? on the straight line ?? + ?? = ?? is- (A) 4 (B) ?? v ?? (C) 8 (D) ?? v ?? Ans: (C) ?? = 1 - ?? ? ?? 2 = 4 - 4?? ? ?? 2 + 4?? - 4 = 0 ( ?? 2 - ?? 1 ) 2 = ( 4) 2 + 4 × 4 = 32 ? |?? 2 - ?? 1 | = 4v 2 |?? 2 - ?? 1 | = ( 1 - ?? 2 )- ( 1 - ?? 1 ) = |?? 2 - ?? 1 | ? Length of chord = v 2 × |?? 2 - ?? 1 | = 8 Q2: A parabola is drawn with its focus at ( ?? , ?? ) and vertex at the focus of the parabola ?? ?? - ???? ?? - ?? ?? + ?? = 0 . The equation of the parabola is - (A) ?? ?? - ?? ?? - ?? ?? + ???? = ?? (B) ?? ?? - ?? ?? - ?? ?? + ???? = ?? (C) ?? ?? - ?? ?? + ?? ?? - ???? = ?? (D) ?? ?? + ?? ?? - ?? ?? - ???? = ?? Ans: (A) ?? 2 - 12?? - 4?? + 4 = 0 ( ?? - 2) 2 = 12?? Let ?? = ?? and ?? = ?? + 2 ? ?? 2 = 12?? Focus of parabola is ( 3,0) ? Focus in original coordinate system is ( 3,2) ? Vertex of new parabola is ( 3,2) &focus = ( 3,4) ? ?? = 2 Let ?? = ?? + 3&?? = ?? + 2 ? Equation is ?? 2 = 8?? ? ( ?? - 3) 2 = 8( ?? - 2) ? ?? 2 - 6?? - 8?? + 25 = 0 is the equation of parabola Q3: The curve describes parametrically by ?? = ?? ?? - ?? ?? + ?? , ?? = ?? ?? + ???? + ?? represents- (A) Straight line (B) Pair of straight line (C) Circle (D) Parabola Ans: (D) ?? = ?? 2 - 2?? + 2, ?? = ?? 2 + 2?? + 2 ?? + ?? - 4 = 2?? 2 ?? - ?? = 4?? ? ( ?? + ?? - 4, ?? - ?? ) lies on a parabola ? ( ?? + ?? , ?? - ?? ) lies on a parabola by rotating axis ( ?? , ?? ) lies on parabola Q4: If ?? = ?? ?? - ?? is a tangent to the parabola ?? ?? = ?? ?? (?? - ?? ?? ), then 'a' is equal to- (A) 1 (B) -1 (C) ???? ?? (D) -???? ?? Ans: (D) ?? 2 = 4?? (?? - 1 3 ) Let ?? = ?? and ?? = ?? + 1 3 ? Equation of parabola is ?? 2 = 4???? ? equation of line is ?? = 2?? - 7 3 ?? = ?? ?? ? - 7 3 = ?? 2 ? ?? = - 14 3 Q5: Two tangents to the parabola ?? ?? = ?? a make angles ?? ?? and ?? ?? with the ?? -axis. The locus of their point of intersection if ?????? ?? ?? ?????? ?? ?? = ?? is- (A) ?? ?? ?? = ?????? (B) ?? ?? ?? = ?????? (C) ?? ?? = ?? ???? (D) None Page 3 JEE Solved Example on Parabola JEE Mains Q1: The length of the chord intercepted by the parabola ?? ?? = ?? ?? on the straight line ?? + ?? = ?? is- (A) 4 (B) ?? v ?? (C) 8 (D) ?? v ?? Ans: (C) ?? = 1 - ?? ? ?? 2 = 4 - 4?? ? ?? 2 + 4?? - 4 = 0 ( ?? 2 - ?? 1 ) 2 = ( 4) 2 + 4 × 4 = 32 ? |?? 2 - ?? 1 | = 4v 2 |?? 2 - ?? 1 | = ( 1 - ?? 2 )- ( 1 - ?? 1 ) = |?? 2 - ?? 1 | ? Length of chord = v 2 × |?? 2 - ?? 1 | = 8 Q2: A parabola is drawn with its focus at ( ?? , ?? ) and vertex at the focus of the parabola ?? ?? - ???? ?? - ?? ?? + ?? = 0 . The equation of the parabola is - (A) ?? ?? - ?? ?? - ?? ?? + ???? = ?? (B) ?? ?? - ?? ?? - ?? ?? + ???? = ?? (C) ?? ?? - ?? ?? + ?? ?? - ???? = ?? (D) ?? ?? + ?? ?? - ?? ?? - ???? = ?? Ans: (A) ?? 2 - 12?? - 4?? + 4 = 0 ( ?? - 2) 2 = 12?? Let ?? = ?? and ?? = ?? + 2 ? ?? 2 = 12?? Focus of parabola is ( 3,0) ? Focus in original coordinate system is ( 3,2) ? Vertex of new parabola is ( 3,2) &focus = ( 3,4) ? ?? = 2 Let ?? = ?? + 3&?? = ?? + 2 ? Equation is ?? 2 = 8?? ? ( ?? - 3) 2 = 8( ?? - 2) ? ?? 2 - 6?? - 8?? + 25 = 0 is the equation of parabola Q3: The curve describes parametrically by ?? = ?? ?? - ?? ?? + ?? , ?? = ?? ?? + ???? + ?? represents- (A) Straight line (B) Pair of straight line (C) Circle (D) Parabola Ans: (D) ?? = ?? 2 - 2?? + 2, ?? = ?? 2 + 2?? + 2 ?? + ?? - 4 = 2?? 2 ?? - ?? = 4?? ? ( ?? + ?? - 4, ?? - ?? ) lies on a parabola ? ( ?? + ?? , ?? - ?? ) lies on a parabola by rotating axis ( ?? , ?? ) lies on parabola Q4: If ?? = ?? ?? - ?? is a tangent to the parabola ?? ?? = ?? ?? (?? - ?? ?? ), then 'a' is equal to- (A) 1 (B) -1 (C) ???? ?? (D) -???? ?? Ans: (D) ?? 2 = 4?? (?? - 1 3 ) Let ?? = ?? and ?? = ?? + 1 3 ? Equation of parabola is ?? 2 = 4???? ? equation of line is ?? = 2?? - 7 3 ?? = ?? ?? ? - 7 3 = ?? 2 ? ?? = - 14 3 Q5: Two tangents to the parabola ?? ?? = ?? a make angles ?? ?? and ?? ?? with the ?? -axis. The locus of their point of intersection if ?????? ?? ?? ?????? ?? ?? = ?? is- (A) ?? ?? ?? = ?????? (B) ?? ?? ?? = ?????? (C) ?? ?? = ?? ???? (D) None Ans: ( ?? ) cota 1 = 2cota 2 ? tana 1 = tan ?? 2 2 let ?? = ( ?? ?? 1 2 , 2?? ?? 1 ) &?? = ( ?? ?? 2 2 , 2?? ?? 2 ) ?? = point of intersection = ( ?? ?? 1 ?? 2 , ?? ( ?? 1 + ?? 2 ) ) = ( ?? , ?? ) Slope of ?? = 1 ?? 1 and slope of ?? = 1 ?? 2 ? 1 ?? 1 = 1 2?? 2 ? ?? 1 = 2?? 2 ? ?? = 2?? ?? 2 2 ?? = 3?? ?? 2 ? ( ?? 3?? ) 2 × 2?? = ?? ? 2?? 2 = 9???? is locus of ?? Q6: Through the vertex ' ?? ' of the parabola ?? ?? = ?? ???? , variable chords ???? and ???? are drawn at right angles. If the variable chord ???? intersects the axis of ?? at ?? , then distance OR- (A) Varies with different positions of ?? and ?? (B) Equals the semi latus rectum of the parabola (C) Equals latus rectum of the parabola (D) Equals double the latus rectum of the parabola Ans: (C) Let P be ( ?? ?? 1 2 , 2?? ?? 1 ) and ?? be ( ?? ?? 2 2 , 2?? ?? 2 ) OP ? OQ ? 2t 1 t 1 2 × 2t 2 t 2 2 = -1 ? t 2 = - 4 t 1 The equation of PQ is ( ?? - ?? ?? 1 ) = 2 ?? 1 - 4 ?? 1 ( ?? - ?? ?? 1 2 ) ?? ?? = (?? 1 - 4 ?? 1 ) × -2???? 2 + ?? ?? 1 2 = 4?? ? ?? = ( 0,4?? ) ? ???? = 4?? Page 4 JEE Solved Example on Parabola JEE Mains Q1: The length of the chord intercepted by the parabola ?? ?? = ?? ?? on the straight line ?? + ?? = ?? is- (A) 4 (B) ?? v ?? (C) 8 (D) ?? v ?? Ans: (C) ?? = 1 - ?? ? ?? 2 = 4 - 4?? ? ?? 2 + 4?? - 4 = 0 ( ?? 2 - ?? 1 ) 2 = ( 4) 2 + 4 × 4 = 32 ? |?? 2 - ?? 1 | = 4v 2 |?? 2 - ?? 1 | = ( 1 - ?? 2 )- ( 1 - ?? 1 ) = |?? 2 - ?? 1 | ? Length of chord = v 2 × |?? 2 - ?? 1 | = 8 Q2: A parabola is drawn with its focus at ( ?? , ?? ) and vertex at the focus of the parabola ?? ?? - ???? ?? - ?? ?? + ?? = 0 . The equation of the parabola is - (A) ?? ?? - ?? ?? - ?? ?? + ???? = ?? (B) ?? ?? - ?? ?? - ?? ?? + ???? = ?? (C) ?? ?? - ?? ?? + ?? ?? - ???? = ?? (D) ?? ?? + ?? ?? - ?? ?? - ???? = ?? Ans: (A) ?? 2 - 12?? - 4?? + 4 = 0 ( ?? - 2) 2 = 12?? Let ?? = ?? and ?? = ?? + 2 ? ?? 2 = 12?? Focus of parabola is ( 3,0) ? Focus in original coordinate system is ( 3,2) ? Vertex of new parabola is ( 3,2) &focus = ( 3,4) ? ?? = 2 Let ?? = ?? + 3&?? = ?? + 2 ? Equation is ?? 2 = 8?? ? ( ?? - 3) 2 = 8( ?? - 2) ? ?? 2 - 6?? - 8?? + 25 = 0 is the equation of parabola Q3: The curve describes parametrically by ?? = ?? ?? - ?? ?? + ?? , ?? = ?? ?? + ???? + ?? represents- (A) Straight line (B) Pair of straight line (C) Circle (D) Parabola Ans: (D) ?? = ?? 2 - 2?? + 2, ?? = ?? 2 + 2?? + 2 ?? + ?? - 4 = 2?? 2 ?? - ?? = 4?? ? ( ?? + ?? - 4, ?? - ?? ) lies on a parabola ? ( ?? + ?? , ?? - ?? ) lies on a parabola by rotating axis ( ?? , ?? ) lies on parabola Q4: If ?? = ?? ?? - ?? is a tangent to the parabola ?? ?? = ?? ?? (?? - ?? ?? ), then 'a' is equal to- (A) 1 (B) -1 (C) ???? ?? (D) -???? ?? Ans: (D) ?? 2 = 4?? (?? - 1 3 ) Let ?? = ?? and ?? = ?? + 1 3 ? Equation of parabola is ?? 2 = 4???? ? equation of line is ?? = 2?? - 7 3 ?? = ?? ?? ? - 7 3 = ?? 2 ? ?? = - 14 3 Q5: Two tangents to the parabola ?? ?? = ?? a make angles ?? ?? and ?? ?? with the ?? -axis. The locus of their point of intersection if ?????? ?? ?? ?????? ?? ?? = ?? is- (A) ?? ?? ?? = ?????? (B) ?? ?? ?? = ?????? (C) ?? ?? = ?? ???? (D) None Ans: ( ?? ) cota 1 = 2cota 2 ? tana 1 = tan ?? 2 2 let ?? = ( ?? ?? 1 2 , 2?? ?? 1 ) &?? = ( ?? ?? 2 2 , 2?? ?? 2 ) ?? = point of intersection = ( ?? ?? 1 ?? 2 , ?? ( ?? 1 + ?? 2 ) ) = ( ?? , ?? ) Slope of ?? = 1 ?? 1 and slope of ?? = 1 ?? 2 ? 1 ?? 1 = 1 2?? 2 ? ?? 1 = 2?? 2 ? ?? = 2?? ?? 2 2 ?? = 3?? ?? 2 ? ( ?? 3?? ) 2 × 2?? = ?? ? 2?? 2 = 9???? is locus of ?? Q6: Through the vertex ' ?? ' of the parabola ?? ?? = ?? ???? , variable chords ???? and ???? are drawn at right angles. If the variable chord ???? intersects the axis of ?? at ?? , then distance OR- (A) Varies with different positions of ?? and ?? (B) Equals the semi latus rectum of the parabola (C) Equals latus rectum of the parabola (D) Equals double the latus rectum of the parabola Ans: (C) Let P be ( ?? ?? 1 2 , 2?? ?? 1 ) and ?? be ( ?? ?? 2 2 , 2?? ?? 2 ) OP ? OQ ? 2t 1 t 1 2 × 2t 2 t 2 2 = -1 ? t 2 = - 4 t 1 The equation of PQ is ( ?? - ?? ?? 1 ) = 2 ?? 1 - 4 ?? 1 ( ?? - ?? ?? 1 2 ) ?? ?? = (?? 1 - 4 ?? 1 ) × -2???? 2 + ?? ?? 1 2 = 4?? ? ?? = ( 0,4?? ) ? ???? = 4?? Q7: A point ?? moves such that the difference between its distances from the origin and from the axis of ' ?? ' is always a constant ?? . The locus- (A) A straight line having equal intercepts ?? on the axis (B) A circle having its centre at (?? , - ?? ?? ) & passing through (?? v ?? , ?? ?? ) (C) A parabola with its vertex at (?? , - ?? ?? ) & passing through (?? v ?? , ?? ?? ) (D) None of these Ans: (C) v?? 2 + ?? 2 - ?? = ?? ? ?? 2 + ?? 2 = ?? 2 + 2???? + ?? 2 ?? 2 = 2?? (?? + ?? 2 ) ? Vertex is (0, - ?? 2 ) It passes through (?? v 2, ?? 2 ) Q8: Tangents are drawn from the point ( -?? , ?? ) on the parabola ?? ?? = ?? ?? . The length, these tangents will intercept on the line ?? = ?? , is- (A) 6 (B) ?? v ?? (C) ?? v ?? (D) None Ans: (B) Equation of tangent is ?? = ???? + 1 ?? ( -1,2) lies on it, then 2 = -?? + 1 ?? ? ?? 2 + 2?? - 1 = 0 ?? = 2, so ?? 1 - ?? 2 = 2( ?? 1 - ?? 2 )+ ?? 2 - ?? 1 ?? 1 ?? 2 ? ( ?? - ?? 2 ) 2 = ( ?? 1 - ?? 2 ) 2 (2 - 1 ?? 1 ?? 2 ) 2 = [( ?? 1 + ?? 2 ) 2 - 4?? 1 ?? 2 ] (2 - 1 ?? 1 ?? 2 ) 2 = [( ?? 1 + ?? 2 ) 2 - 4?? 1 ?? 2 ] (2 - 1 ?? 1 ?? 2 ) 2 ( ?? 1 - ?? 2 ) 2 = ( 4 + 4) ( 2 + 1) 2 = 8 × 9 ? |?? 1 - ?? 2 | = 6v 2 Page 5 JEE Solved Example on Parabola JEE Mains Q1: The length of the chord intercepted by the parabola ?? ?? = ?? ?? on the straight line ?? + ?? = ?? is- (A) 4 (B) ?? v ?? (C) 8 (D) ?? v ?? Ans: (C) ?? = 1 - ?? ? ?? 2 = 4 - 4?? ? ?? 2 + 4?? - 4 = 0 ( ?? 2 - ?? 1 ) 2 = ( 4) 2 + 4 × 4 = 32 ? |?? 2 - ?? 1 | = 4v 2 |?? 2 - ?? 1 | = ( 1 - ?? 2 )- ( 1 - ?? 1 ) = |?? 2 - ?? 1 | ? Length of chord = v 2 × |?? 2 - ?? 1 | = 8 Q2: A parabola is drawn with its focus at ( ?? , ?? ) and vertex at the focus of the parabola ?? ?? - ???? ?? - ?? ?? + ?? = 0 . The equation of the parabola is - (A) ?? ?? - ?? ?? - ?? ?? + ???? = ?? (B) ?? ?? - ?? ?? - ?? ?? + ???? = ?? (C) ?? ?? - ?? ?? + ?? ?? - ???? = ?? (D) ?? ?? + ?? ?? - ?? ?? - ???? = ?? Ans: (A) ?? 2 - 12?? - 4?? + 4 = 0 ( ?? - 2) 2 = 12?? Let ?? = ?? and ?? = ?? + 2 ? ?? 2 = 12?? Focus of parabola is ( 3,0) ? Focus in original coordinate system is ( 3,2) ? Vertex of new parabola is ( 3,2) &focus = ( 3,4) ? ?? = 2 Let ?? = ?? + 3&?? = ?? + 2 ? Equation is ?? 2 = 8?? ? ( ?? - 3) 2 = 8( ?? - 2) ? ?? 2 - 6?? - 8?? + 25 = 0 is the equation of parabola Q3: The curve describes parametrically by ?? = ?? ?? - ?? ?? + ?? , ?? = ?? ?? + ???? + ?? represents- (A) Straight line (B) Pair of straight line (C) Circle (D) Parabola Ans: (D) ?? = ?? 2 - 2?? + 2, ?? = ?? 2 + 2?? + 2 ?? + ?? - 4 = 2?? 2 ?? - ?? = 4?? ? ( ?? + ?? - 4, ?? - ?? ) lies on a parabola ? ( ?? + ?? , ?? - ?? ) lies on a parabola by rotating axis ( ?? , ?? ) lies on parabola Q4: If ?? = ?? ?? - ?? is a tangent to the parabola ?? ?? = ?? ?? (?? - ?? ?? ), then 'a' is equal to- (A) 1 (B) -1 (C) ???? ?? (D) -???? ?? Ans: (D) ?? 2 = 4?? (?? - 1 3 ) Let ?? = ?? and ?? = ?? + 1 3 ? Equation of parabola is ?? 2 = 4???? ? equation of line is ?? = 2?? - 7 3 ?? = ?? ?? ? - 7 3 = ?? 2 ? ?? = - 14 3 Q5: Two tangents to the parabola ?? ?? = ?? a make angles ?? ?? and ?? ?? with the ?? -axis. The locus of their point of intersection if ?????? ?? ?? ?????? ?? ?? = ?? is- (A) ?? ?? ?? = ?????? (B) ?? ?? ?? = ?????? (C) ?? ?? = ?? ???? (D) None Ans: ( ?? ) cota 1 = 2cota 2 ? tana 1 = tan ?? 2 2 let ?? = ( ?? ?? 1 2 , 2?? ?? 1 ) &?? = ( ?? ?? 2 2 , 2?? ?? 2 ) ?? = point of intersection = ( ?? ?? 1 ?? 2 , ?? ( ?? 1 + ?? 2 ) ) = ( ?? , ?? ) Slope of ?? = 1 ?? 1 and slope of ?? = 1 ?? 2 ? 1 ?? 1 = 1 2?? 2 ? ?? 1 = 2?? 2 ? ?? = 2?? ?? 2 2 ?? = 3?? ?? 2 ? ( ?? 3?? ) 2 × 2?? = ?? ? 2?? 2 = 9???? is locus of ?? Q6: Through the vertex ' ?? ' of the parabola ?? ?? = ?? ???? , variable chords ???? and ???? are drawn at right angles. If the variable chord ???? intersects the axis of ?? at ?? , then distance OR- (A) Varies with different positions of ?? and ?? (B) Equals the semi latus rectum of the parabola (C) Equals latus rectum of the parabola (D) Equals double the latus rectum of the parabola Ans: (C) Let P be ( ?? ?? 1 2 , 2?? ?? 1 ) and ?? be ( ?? ?? 2 2 , 2?? ?? 2 ) OP ? OQ ? 2t 1 t 1 2 × 2t 2 t 2 2 = -1 ? t 2 = - 4 t 1 The equation of PQ is ( ?? - ?? ?? 1 ) = 2 ?? 1 - 4 ?? 1 ( ?? - ?? ?? 1 2 ) ?? ?? = (?? 1 - 4 ?? 1 ) × -2???? 2 + ?? ?? 1 2 = 4?? ? ?? = ( 0,4?? ) ? ???? = 4?? Q7: A point ?? moves such that the difference between its distances from the origin and from the axis of ' ?? ' is always a constant ?? . The locus- (A) A straight line having equal intercepts ?? on the axis (B) A circle having its centre at (?? , - ?? ?? ) & passing through (?? v ?? , ?? ?? ) (C) A parabola with its vertex at (?? , - ?? ?? ) & passing through (?? v ?? , ?? ?? ) (D) None of these Ans: (C) v?? 2 + ?? 2 - ?? = ?? ? ?? 2 + ?? 2 = ?? 2 + 2???? + ?? 2 ?? 2 = 2?? (?? + ?? 2 ) ? Vertex is (0, - ?? 2 ) It passes through (?? v 2, ?? 2 ) Q8: Tangents are drawn from the point ( -?? , ?? ) on the parabola ?? ?? = ?? ?? . The length, these tangents will intercept on the line ?? = ?? , is- (A) 6 (B) ?? v ?? (C) ?? v ?? (D) None Ans: (B) Equation of tangent is ?? = ???? + 1 ?? ( -1,2) lies on it, then 2 = -?? + 1 ?? ? ?? 2 + 2?? - 1 = 0 ?? = 2, so ?? 1 - ?? 2 = 2( ?? 1 - ?? 2 )+ ?? 2 - ?? 1 ?? 1 ?? 2 ? ( ?? - ?? 2 ) 2 = ( ?? 1 - ?? 2 ) 2 (2 - 1 ?? 1 ?? 2 ) 2 = [( ?? 1 + ?? 2 ) 2 - 4?? 1 ?? 2 ] (2 - 1 ?? 1 ?? 2 ) 2 = [( ?? 1 + ?? 2 ) 2 - 4?? 1 ?? 2 ] (2 - 1 ?? 1 ?? 2 ) 2 ( ?? 1 - ?? 2 ) 2 = ( 4 + 4) ( 2 + 1) 2 = 8 × 9 ? |?? 1 - ?? 2 | = 6v 2 Q9: Which one of the following equations represented parametric -cally represents equation to a parabolic profile? (A) ?? = ???????? ?? ; ?? = ???????? ?? (B) ?? ?? - ?? = -?????? ?? ; ?? = ?? ?????? ?? ?? ?? (C) v ?? = ?????? ?? ; v?? = ?????? ?? (D) ?? = v ?? - ?????? ?? ; ?? = ?????? ?? + ?????? ?? ?? Ans: (B) (A) is an ellipse (B) ?? 2 = 2 - cos ?? = 2 - (2cos 2 ?? 2 - 1) = 2 - ( ?? 2 - 1) ? ?? 2 = 1 2 ( 6 - ?? ) ? It is a parabola (C) is a line in 1 st quadrant (D) ?? = v 1 - sin ?? , ?? = sin ?? 2 + cos ?? 2 ? ?? 2 = sin 2 ?? 2 + cos 2 ?? 2 + 2sin ?? 2 cos ?? 2 = 1 + sin ?? = 1 + 1 - ?? 2 (A circle) Q10: From an external point ?? , pair of tangent lines are drawn to the parabola, ?? ?? = ???? . If ?? ?? and ?? ?? are the inclinations of these tangents with the axis of ?? such that, ?? ?? + ?? ?? = ?? ?? , then the locus of ?? is- (A) ?? - ?? + ?? = ?? (B) ?? + ?? - ?? = ?? (C) ?? - ?? - ?? = ?? (D) ?? + ?? + ?? = ?? Ans: (C) Let A ( ?? ?? 1 2 , 2?? ?? 2 ) &?? ( ?? ?? 2 2 , 2?? ?? 2 ) be points on parabola. P = ( ?? ?? 1 t 2 '?? ( t 1 + t 2 ) ) = ( x, y) slope of ???? is 1 ?? 1 & slope of ???? is 1 ?? 2 tan ( ?? 1 + ?? 2 )= 1 ? 1 t 1 + 1 t 2 1 - 1 t 1 t 2 = 1 ? t 1 + t 2 t 1 t 2 - 1 = 1 ? ?? ?? - 1 = 1Read More
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