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The center of the ellipse \(\dfrac{(x+y2^2)}{9}+\dfrac{(xy)^2}{16}=1 \) is________
The center of the given ellipse is the point of intersection of the lines and . After solving above two equations we get and .
Centere
The parametric coordinate of any of the parabola \( y^2 =4ax \) is_______
( Equation of Parabola)
The parametric form of this parabola is
Coordinate of any points is
The equation of parabola with vertex at origin the axis is along \( x \)axis and passing through the point \( (2, 3 ) \) is_____
Equation of the parabola is (i)
Given point
Putting the value of in (i).
At what point of the parabola \(x^2=ay \) is the abscissa three times that of ordinate ______
Given, Parabola is
Let be the point on the parabola.
So ( According to question )
Since lies on the parabola .
From above two equations we get
When
When
In an ellipse the distance between its foci is \( 6 \) and its minor axis is \( 8 \) then \( e \) is ________
Given, distance between foci
Minor axis
Now,
The equation of parabola whose focus is \( (3 , 0) \) and directrix is \( 3x+4y =1 \)______
Given, , Equation of directrix
Let be any point on parabola and be the length of perpendicular.
The parametric representation of \( ( 2+t^2 , 2t+1) \) represents_____
(i)
(ii)
From (i) and (ii)
This represents the equation of parabola.
The equation of a hyperbola with foci on the \( x \)axis is_____
Equation of hyperbola is
The line \( lx+my+n=0 \) will touch the parabola \( y^2=4ax \) if_______
Given,
This will touch the parabola if
A rod of length \( 12\operatorname {cm} \) moves with its and always touching coordinate axes. Then the equation of the locus of a point \( P \) on the rod which \( 3 \operatorname {cm} \) from the end in contact with the \( x \)axis is_______
Hence
Now
Since
Now in
Again in
From above two equations , we get
If \( (a, b) \) is the midpoint of a chord passing through the vertex of the parabola \( y^2=4x \), then______
Let be the other end of the chord of .
Now,
And
The equation of parabola with vertex \( (2, 1 ) \) and focus \((2 ,4) \) is______
Given, vertex , Focus
Parabola axis of symmetry as
Hence, the equation of the parabola is of the type where is vertex.
Now focus
So, focus
Now equation of parabola is
If a parabolic reflector is \( 20 \operatorname {cm} \) in diameter and \(5 \operatorname {cm} \) deep then the focus of parabolic reflector is_______
Given, diameter of the parabola is .
Equation of parabola is
Parabola passes through the point
So, focus
The parametric form of the ellipse \(4(x+1)^2+ (y1)^2=4 \) is_______
Given,
Parametric equation of given ellipse is and
A man running a race course notes that the sum of the distance from the two flag posts from him is always \( 10 \operatorname {meter} \) and the distance between the flag posts is \(8 \operatorname { meter} \). The equation of the posts traced by the man is_________
Let the equation of ellipse is
Given,
Given,
Now,
Equation of the ellipse is
Let \( O \) be the vertex and \( Q \) be any point on the parabola \(x^2=8y \). If the point \( P \) divides the line segment \(OQ \) internalyin ratio \( 1:3 \) , then the locus of \( P \) is_______
Given,
Let
Let divides joining and in ratio .
and
, required locus.
The locus of the midpoint of the line segment joining the focus to a moving point on the parabola \( y^2=4ax \) is another parabola with directrix, is_______________
Let be midpoint of the line segment joining the focus and . Then,
Given,
Locus of is
Its directrix is
The equation of the directrix of the parabola \( y^2+4y+4x+2=0 \) is_______
Given,
Replacing
We have,
This is the parabola with directrix at
If the line \(x1=0 \) is the directrix of the parabola \(y^2kx+8=0 \), then value of \( k \) is_______
Given,
Shifting the origin, where
Directrix of standard parabola is
Directrix of original parabola is
Now, also coincides with on solving this we get .
The radius of a circle having minimum area, which touches the curve \( y=4x^2 \) and the lines \( y=x \) is_______
Let radius of circle is .
Coordinate of centre is
Since, circle touches the line in quadrant.
But
The slope of the line touching both the parabolas \( y^2=4x \) and \(x^2=32y \) is________
Let the tangent to the parabola be , if it touches the curve, then , to get the value of .
For parabola , .
Let be tangent line and it touches the parabola
The tangent at \( (1 ,7) \) to the curves \( x^2=y6x \) touches the circle \(x^2+y^2+16x+12y+c=0 \) at_______
The tangent at to the parabola is [ replacing mathjax] and ]
Which is also tangent to the circle
i.e.,
It must have two equal roots ()
and
Point of contact is
The angle between the tangents drawn from the point \((1 , 4) \) to the parabola \( y^2=4x \) is________
Tangent to is
Given , tangent
Since, tangent passes through
and
The focal chord to \( y^2=16x \) is tangent to \( (x6)^2+y^2=2 \), then the possible values of the slope of this chord are_______
Given, and
Now, tangents are drawn from to
Since, is tangent to circle.
Slope of the focal chord as tangent to circle
The equation of the common tangent to the curves \( y^2=8x \) and \( xy=1 \) is_______
Tangent to the curve is . So it must satisfy .
Since
Equation of common tanget is