Maths-
General
Easy
Question
If P (q) and Q
are two points on the ellipse
, then locus of the mid – point of PQ is
- none of these
Hint:
Equation of ellipse = ![fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1](data:image/png;base64,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)
The correct answer is: ![fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals fraction numerator 1 over denominator 2 end fraction](data:image/png;base64,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)
Coordinate of Point P(
) = (a cos
, b sin
)
Coordinate of Point Q (
) = ( a cos(
) ,
)
Coordinate of Point Q (
) = ( -a sin(
) ,
)
Let R be the midpoint of P and Q and coordinates of R = (h, k)
![h space equals space fraction numerator a cos theta space minus space a sin theta over denominator 2 end fraction space a n d space k space equals space fraction numerator b sin theta space plus space b cos theta over denominator 2 end fraction](data:image/png;base64,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)
![fraction numerator 2 h over denominator a end fraction space equals space cos theta space minus sin theta space a n d space fraction numerator 2 k over denominator b end fraction space equals space sin theta space plus space cos theta](data:image/png;base64,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)
![left parenthesis fraction numerator 2 h over denominator a end fraction right parenthesis squared space plus space left parenthesis fraction numerator 2 k over denominator b end fraction right parenthesis squared space equals space open parentheses cos theta space minus sin theta close parentheses squared space plus left parenthesis cos theta space plus sin theta right parenthesis squared](data:image/png;base64,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)
![left parenthesis fraction numerator 4 h squared over denominator a squared end fraction right parenthesis space plus space left parenthesis fraction numerator 4 k squared over denominator b squared end fraction right parenthesis space equals space open parentheses cos squared theta space plus sin squared theta space minus space 2 cos theta sin theta close parentheses space plus left parenthesis cos squared theta space plus sin squared theta space plus space 2 cos theta sin theta right parenthesis](data:image/png;base64,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)
![left parenthesis fraction numerator 4 h squared over denominator a squared end fraction right parenthesis space plus space left parenthesis fraction numerator 4 k squared over denominator b squared end fraction right parenthesis space equals space 2](data:image/png;base64,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)
comparing with equation of ellipse i.e. ![fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1](data:image/png;base64,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)
![x squared over a squared plus y squared over b squared equals 1 half](data:image/png;base64,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)
Related Questions to study
Maths-
If CP and CD are semi – conjugate diameters of an ellipse
then CP2 + CD2 =
If CP and CD are semi – conjugate diameters of an ellipse
then CP2 + CD2 =
Maths-General
Maths-
The normals to the curve x = a (
+ sin
), y = a (1 – cos
) at the points
= (2n + 1)
, n
I are all -
The normals to the curve x = a (
+ sin
), y = a (1 – cos
) at the points
= (2n + 1)
, n
I are all -
Maths-General
Maths-
The normal to the curve x = 3 cos
–
, y = 3 sin
– ![sin cubed](data:image/png;base64,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)
at the point
=
/4 passes through the point -
The normal to the curve x = 3 cos
–
, y = 3 sin
– ![sin cubed](data:image/png;base64,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)
at the point
=
/4 passes through the point -
Maths-General
Maths-
The normal of the curve given by the equation x = a (sin
+ cos
), y = a (sin
– cos
) at the point Q is -
The normal of the curve given by the equation x = a (sin
+ cos
), y = a (sin
– cos
) at the point Q is -
Maths-General
Maths-
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 -
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 -
Maths-General
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