Maths-
General
Easy
Question
If
= 1 is a tangent to the curve x = 4t,y =
, t
R then -
- a > 0 , b > 0
- a > 0, b < 0
- a < 0, b > 0
- None of these
Hint:
Find slope i.e ![fraction numerator d y over denominator d x end fraction](data:image/png;base64,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)
The correct answer is: a > 0 , b > 0
Given : x = 4t, y =
, t
R
, ( the curve is a rectangular parabola)
![rightwards double arrow fraction numerator x y 1 space plus space y x 1 over denominator 2 end fraction space equals space k squared
rightwards double arrow x y 1 space plus y x 1 space equals space 2 k squared](data:image/png;base64,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)
![rightwards double arrow space L e t space x 1 space equals space k t space a n d space y 1 space equals space k over t
rightwards double arrow x cross times k over t space plus space y space cross times k t space equals space 2 k squared
rightwards double arrow x over t space plus space t y space space equals space 2 k
rightwards double arrow S l o p e space o f space tan g e n t space equals space fraction numerator negative space c o e f f e c i e n t space o f space x over denominator c o e f f e c i e n t space o f space y end fraction
rightwards double arrow S l o p e space o f space tan g e n t space equals space fraction numerator begin display style bevelled fraction numerator negative 1 over denominator t end fraction end style over denominator t end fraction space equals fraction numerator negative 1 over denominator t squared end fraction less than space 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![rightwards double arrow x over a space plus space y over b space equals space 1 comma space i. e. space tan g e n t space f o r space x y space equals space k squared
rightwards double arrow S l o p e space o f space tan g e n t space less than space 0
rightwards double arrow S l o p e space o f space l i n e space equals space fraction numerator begin display style bevelled fraction numerator negative 1 over denominator a end fraction end style over denominator begin display style bevelled 1 over b end style end fraction space less than space 0
rightwards double arrow S l o p e space o f space l i n e space equals space fraction numerator negative b over denominator a end fraction less than space 0
rightwards double arrow S l o p e space o f space l i n e space equals space b over a space greater than space 0
rightwards double arrow T h i s space i m p l i e s space e i t h e r space b space greater than space 0 space comma space a greater than space 0 space o r space b space less than 0 space space a n d space a less than space 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![fraction numerator d y over denominator d x end fraction space equals space fraction numerator begin display style bevelled fraction numerator d y over denominator d t end fraction end style over denominator begin display style bevelled fraction numerator d x over denominator d t end fraction end style end fraction
fraction numerator d y over denominator d t end fraction space equals space 4 space a n d space space fraction numerator d x over denominator d t end fraction space equals negative space 4 over t squared
S u b s t i t u t i n g space t h e space v a l u e s
fraction numerator d y over denominator d x end fraction space equals space fraction numerator begin display style bevelled fraction numerator d y over denominator d t end fraction end style over denominator begin display style bevelled fraction numerator d x over denominator d t end fraction end style end fraction space equals space fraction numerator 4 over denominator begin display style bevelled fraction numerator negative 4 over denominator t squared end fraction end style end fraction space equals space minus 1 over t squared space](data:image/png;base64,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The line
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The line
+
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An object moves in a straight line. It starts from the rest and its acceleration is . 2 ms2.After reaching a certain point it comes back to the original point. In this movement its acceleration is -3 ms2, till it comes to rest. The total time taken for the movement is 5 second. Calculate the maximum velocity.
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The sum of distances of any point on the ellipse 3 x2 + 4y2 = 24 from its foci is
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The equations x = a
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The line y = 2x + c touches the ellipse
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The eccentricity of the conic 3x2 + 4y2 = 24 is
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The equation of the ellipse whose focus is (1, -1). directrix x – y – 3 = 0 and eccentricity
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The equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents an ellipse if
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On a new year day every student of a class sends a card to every other student. The postman delivers 600 cards. The number of students in the class are :
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A tangent to the ellipse x2 + 4y2 = 4 meets the ellipse x2 + 2y2 = 6 at P and Q. The angle between the tangents at P and Q of the ellipse x2 + 2y2 = 6 is
A tangent to the ellipse x2 + 4y2 = 4 meets the ellipse x2 + 2y2 = 6 at P and Q. The angle between the tangents at P and Q of the ellipse x2 + 2y2 = 6 is
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An ellipse has OB as a semi – minor axis, F, F' are its foci and the angle FBF' is a right angle. Then the eccentricity of the ellipse is
An ellipse has OB as a semi – minor axis, F, F' are its foci and the angle FBF' is a right angle. Then the eccentricity of the ellipse is
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The equation of the tangents drawn at the ends of the major axis of the ellipse 9x2 + 5y2 – 30y = 0 is
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