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The equation of the common tanget touching the circle \( (x3)^2+y^2=9 \) and the parabola \( y^2=4x \) above the \( x \)axis is_______
Tangent : and this touches the circle .
If
Tangent touches the parabola and circle above the axis, then .
Equation of the tangent is
Equation of the common tangent of \(y=x^2, \quad y=x^2+4x4 \) is______
Tangent :
Given
Since
Now
If \( x+y=k \) is normal to \( y^2=12x \) , then \( k \) is_______
is normal to parabola then .
Hence
(Given)
Let \( L \) be a normal to the parabola \( y^2=4x \). If \( L \) passes through the point \((9,6) \), the \( L \) is given by _____
Normal to is which passes through .
Equation of normal is (For )
Find the equation of the normal to the curve \( x^2=4y \) which passes through the point \( (1, 2) \).
Given,
Equation of the normal is and passing through .
The required equation is
Suppose that the normals drawn at three different points on the parabola \( y^2=4x \) passes through the point \((h, 0) \) then ______
Given,
Three different normals are drawn from to .
Then, equation of normals are which passes through
, where
Let \( (x,y) \) be any point on the parabola \( y^2=4x \). Let \( P \) be the point that divides the line segment from \((0,0 ) \) to \( (x,y) \) in the ratio \( 1:3 \). Then, the locus of \( P \) is_____
Let
Hence
is required locus.
The equation of the passing thorugh the foci of the ellipse \( \dfrac{x^2}{16}+\dfrac{y^2}{9}=1 \) and having centre at \( (0,3) \) is_____
Given:
Hence,
Equation of Circle is
If \( P=(x,y), \quad F_1 =(3 , 0), F_2 =(3, 0) \) and \( 16x^2+25y^2=400 \), then \( PF_1+ PF_2 \) is equal to ______
Given,
Hence
Now, foci of the ellipse are
We have,
Now
An ellipse has \( OB \) as a semiminor axis \( F \) and \( F’ \) are foci and angle \( FBF’ \) is \( 90^o \). Then , \( e \) is_______
Since
(Slope of ) (Slope of )
Let \( P \) be a variable point on the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \) with foci \( F_1 \) and \( F_2 \). If \( A \) is the area of the \(\triangle PF_1F_2 \), then the maximum value of \( A \) is ______
Let be any point on the ellipse.
is maximum when
Maximum of
The eccentricity of an ellipse whose centre is at the origin is \( \dfrac{1}{2} \). If one of its directrices is \( x=4 \), then the equation of the normal to it at \( \left( 1 , \dfrac{3}{2} \right)\) is_______
We have and
Now
Equation of the ellipse is
Equation of Normal at is
The area of the quadrilateral formed by the tangents at the end points of the latus rectum to the ellipse \( \dfrac{x^2}{9}+ \dfrac{y^2}{5}=1 \) is_______
Given,
Now
Extremities of one of are
Equation of tangent at is
It intersects and axis at respectively.
Area of equilateral Area of
The line passing through the extremity of the major axis and extremity \( B \) of the minor axis of the ellipse \( x^2+9y^2=9 \) meets its auxilliary circle at \( M \). Then, the area of the triangle with vertices at \( A , M \) and the origin \( O \) is_____
Given, <br.
Equation of the auxilliary
Circle:
Solving above two equations, we get
Now area of
Tangents area drawn to the ellipse \( x^2+2y^2=2 \), then the locus of the midpoint of the intercept made by the tangents between the coordinate axes is_______
Let is on
Equation of tangent: whose intecerpt on coordinate axes are and
Focus of is
Tangents are drawn to the ellipse \( \dfrac{x^2}{27}+y^2=1 \) at \((3 \sqrt{3}\cos \theta, \sin \theta), \quad \theta \left( 0, \dfrac{\pi}{2} \right) \). Then, the value of \( \theta \) such that the sum of intercepts on axes made by this tangent is minimum, is_______
Given, tangent is drawn at to
Equation of tangent is
Thus, Sum of intecepts
For
is minimum.
If \( a > 2b > 0 \), then positive value of \( m \) for which \( y=mxb \sqrt{1+m^2} \) is a common tangent to \( x^2+y^2=b^2 \) and \( (xa)^2+y^2=b^2 \) is______
Given, touches both the circles, so distance from centre equals to radius of both the circles.
The number of values of \( c \) such that the straight line \( y=4x+c \) touches the curve \( \dfrac{x^2}{4}+y^2=1 \) is______
For ellipse, condition of tangency is
Given,
Find the equation to the common tangent in \( 1\text {st} \) quadrant to the circle \( x^2+y^2=16 \) and the ellipse \( \dfrac{x^2}{25}+ \dfrac{y^2}{4}=1 \)
Given, and
Equation of common tangent is
(i)
and (ii)
Since (i) and (ii) are same
Since, tangent is in quadrant
Equation of tangent is
The eccentricity of the hyperbola whose length of the latus rectum is equal to \(8 \) and the length of its conjugate axis is equal to half of its focal distance, is_____
We have,
Consider, a branch of hyperbola \(x^22y^22\sqrt{2}x4\sqrt{2}y6=0 \) with vertex at \( A \). Let \( B \) be one of the end points of its latus rectum. If \( C \) is focus of the hyperbola nearest to \( A \), then the area of \(\triangle ABC \) is (in sq. units)
Given,
(For A)
(For C)
Now,
And
A hyperbola , having the transverse axis of length \(2 \sin \theta \), is confocal with the ellipse \(3x^2+4y^2=12 \), then its equation is_____________
Given,
Equation of hyperbola is
If \( e_1 \) is eccentricity of ellipse \( \dfrac{x^2}{16}+ \dfrac{y^3}{25}=1 \) and \( e_2 \) is eccentricity of hyperbola passing through the foci of the ellipse and \( e_1e_2=1 \), then equation of the hyperbola is________
Given,
Foci of ellipse
Equation of hyperbola is
For hyperbola , \(\dfrac{x^2}{\cos^2 \alpha}\dfrac{y^2}{\sin^2 \alpha}=1 \), which of the following remains constant with change in \(\alpha \)?
Given,
Hence,
Foci
Hence
Vertices are
Hence, foci remains constant with change in
The equation \( \dfrac{x^2}{1r}\dfrac{y^2}{1+r}=1 , \quad r<1 \) represents_______
Given,
is positive and is positive
Given equation is of the form
This represents a hyperbola when .