Maths-
General
Easy
Question
If the tangent at ‘t’ on the curve y =
, x =
meets the curve again at
and is normal to the curve at that point, then value of t must be -
- ±
- ±
- ±
- None of these
Hint:
Slope of tangent = ![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](data:image/png;base64,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)
The correct answer is: ± ![1 third](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAKNJREFUeNpjYMAP1IC4FogvMFAIFgNxGhD/Z6ASGDVo1KBBYdB/LHgUDCbwn0w8CgYLcATibUD8DYh/APEtIJ4AxMKkGrQfiIOAmA1JzACId1DLpZ+oYYgSEN+gxABxIE6EhpMXNbJMAaVeEgTiACA+CcT21AgjHqhhVAE/qGGIHjTASQIHgTgUiFmAmAma0u8DcTypBtkC8RaoV35AU7oPPg0ArQs4erRAu/EAAABidEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mcmFjPjxtbj4xPC9tbj48bW4+MzwvbW4+PC9tZnJhYz48L21hdGg+q+QgMwAAAABJRU5ErkJggg==)
Slope of tangent = ![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](data:image/png;base64,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)
Given : y =
and x = ![4 t squared space plus space 3](data:image/png;base64,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)
![fraction numerator d y over denominator d t end fraction space equals space 24 t squared space a n d space fraction numerator d x over denominator d t end fraction space equals space 8 t](data:image/png;base64,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)
= ![fraction numerator 24 t squared over denominator 8 t end fraction space equals space 3 t](data:image/png;base64,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)
Equation tangent at point 't'
y -y1 = m(x -x1)
![rightwards double arrow y space minus space 8 t cubed space plus space 1 space equals space space 3 t left parenthesis space x minus space 4 t squared space minus 3 right parenthesis
A t space p o i n t space t subscript 1
rightwards double arrow 8 t subscript 1 cubed minus 1 space minus space 8 left parenthesis t right parenthesis cubed space plus space 1 space equals space 3 t left parenthesis 4 t subscript 1 squared space plus space 3 space minus space 4 left parenthesis t right parenthesis squared space minus 3 right parenthesis
rightwards double arrow 8 left parenthesis t subscript 1 cubed space minus space t cubed right parenthesis space equals space 12 t space left parenthesis t subscript 1 squared space minus space t squared right parenthesis
rightwards double arrow 8 left parenthesis t subscript 1 space minus space t right parenthesis left parenthesis t subscript 1 squared space plus space t squared space plus t t subscript 1 right parenthesis space equals space 12 t space left parenthesis space t subscript 1 space minus t right parenthesis left parenthesis t subscript 1 space plus space t right parenthesis
rightwards double arrow 2 t subscript 1 squared space space plus space 2 t squared space plus 2 t subscript 1 t space equals space 3 t t subscript 1 space plus space 3 t squared
rightwards double arrow 2 t subscript 1 squared space minus space t t subscript 1 space minus space t squared space equals space 0
rightwards double arrow 2 t subscript 1 squared space minus 2 space t t subscript 1 space plus space t t subscript 1 space minus space t squared space equals space 0
rightwards double arrow 2 t subscript 1 left parenthesis t subscript 1 space minus t right parenthesis space plus space t left parenthesis t subscript 1 space minus t right parenthesis equals space 0
rightwards double arrow left parenthesis 2 t subscript 1 space end subscript plus space t right parenthesis left parenthesis t subscript 1 space minus t right parenthesis space equals space 0
rightwards double arrow t space equals space t subscript 1 left parenthesis n o t space p o s s i b l e right parenthesis space. space t space equals space minus 2 t subscript 1 space end subscript
rightwards double arrow T h u s comma space t space equals space minus 2 t subscript 1 space end subscript
rightwards double arrow t subscript 1 space end subscript space equals negative space t over 2
rightwards double arrow S l o p e space o f space n o r m a l space equals space minus fraction numerator 1 over denominator 3 t end 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)
Equation of normal
![y space minus space 8 t cubed space plus 1 space equals space minus fraction numerator 1 over denominator 3 t end fraction left parenthesis x space minus space 4 t squared space minus 3 right parenthesis
A t space p o i n t space t subscript 1
rightwards double arrow 8 t subscript 1 cubed minus 1 space minus space 8 left parenthesis t right parenthesis cubed space plus space 1 space equals space space minus fraction numerator 1 over denominator 3 t end fraction left parenthesis 4 t subscript 1 squared space plus space 3 space minus space 4 left parenthesis t right parenthesis squared space minus 3 right parenthesis
rightwards double arrow 8 left parenthesis t subscript 1 to the power of 3 space end exponent space minus space t cubed right parenthesis space equals space minus fraction numerator 4 over denominator 3 t end fraction space left parenthesis space t subscript 1 squared space space minus space t squared right 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![rightwards double arrow 6 t space left parenthesis space t subscript 1 cubed space minus t cubed right parenthesis space equals space left parenthesis t squared space minus space t subscript 1 squared right parenthesis
rightwards double arrow negative space 6 t space space left parenthesis space t squared space plus space t subscript 1 squared space plus t t subscript 1 right parenthesis space equals space t space plus space t subscript 1
S u b s t i t u t e space t h e space v a l u e space o f space t subscript 1
rightwards double arrow negative space 6 t space space left parenthesis space t squared space plus space t over 4 squared space minus space t t over 2 right parenthesis space equals space t space minus t over 2](data:image/png;base64,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)
![fraction numerator rightwards double arrow 6 over denominator 4 end fraction left parenthesis 5 t squared space minus space 2 t squared right parenthesis space equals space 1 half
rightwards double arrow t squared space equals space 1 over 9
rightwards double arrow t space equals space plus-or-minus 1 third](data:image/png;base64,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)
Related Questions to study
Maths-
The tangent at (
,
–
) on the curve y =
–
meets the curve again at Q, then abscissa of Q must be -
The tangent at (
,
–
) on the curve y =
–
meets the curve again at Q, then abscissa of Q must be -
Maths-General
Maths-
If
= 1 is a tangent to the curve x = 4t,y =
, t
R then -
If
= 1 is a tangent to the curve x = 4t,y =
, t
R then -
Maths-General
Maths-
The line
+
= 1 touches the curve
at the point :
The line
+
= 1 touches the curve
at the point :
Maths-General
physics
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.
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.
physicsGeneral
Maths-
For the ellipse
the foci are
For the ellipse
the foci are
Maths-General
Maths-
For the ellipse
, the latus rectum is
For the ellipse
, the latus rectum is
Maths-General
Maths-
The sum of distances of any point on the ellipse 3 x2 + 4y2 = 24 from its foci is
The sum of distances of any point on the ellipse 3 x2 + 4y2 = 24 from its foci is
Maths-General
Maths-
The equations x = a
represent
The equations x = a
represent
Maths-General
Maths-
The equations x = a cos q, y = b sin q, 0 ≤ q < 2 p, a ≠ b, represent
The equations x = a cos q, y = b sin q, 0 ≤ q < 2 p, a ≠ b, represent
Maths-General
Maths-
The line y = 2x + c touches the ellipse
if c is equal to
The line y = 2x + c touches the ellipse
if c is equal to
Maths-General
Maths-
The eccentricity of the conic 3x2 + 4y2 = 24 is
The eccentricity of the conic 3x2 + 4y2 = 24 is
Maths-General
Maths-
The equation of the ellipse whose focus is (1, -1). directrix x – y – 3 = 0 and eccentricity
is
The equation of the ellipse whose focus is (1, -1). directrix x – y – 3 = 0 and eccentricity
is
Maths-General
Maths-
The equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents an ellipse if
The equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents an ellipse if
Maths-General
Maths-
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 :
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 :
Maths-General
Maths-
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
Maths-General