Radius of the circle that passes through origin and touches the parabola $$y^2 = 4ax$$ at the point $$(a, 2a)$$ is
Given equation of parabola is
$$y^{2}=4ax$$
$$\Rightarrow\displaystyle\frac{dy}{dx}=\frac{2a}{y}$$
Slope of tangent at $$(a,2a)$$ = Slope of parabola at $$(a,2a) =1$$
Equation of tangent to parabola at $$(a,2a)$$ is
$$y-2a =(x-a)$$
or,$$y-x-a=0$$
Equation of circle touching the parabola at $$(a,2a)$$ is
$$(x-a)^2+(y-2a)^2+\lambda(y-x-a)=0$$
Since this circle passes through origin
$$ a^2+4a^2-a\lambda = 0$$
$$\Rightarrow \lambda=5a$$
So, the equation of circle is
$$(x-a)^{2}+(y-2a)^{2}+5a(y-x-a)=0$$
$$\Rightarrow x^{2}+y^{2}-7ax+ay=0$$
Comparing with general equation of circle
$$x^2+y^2+2gx+2fy+c=0$$
So, $$\displaystyle g=-\frac{7a}{2},f=\frac{a}{2},c=0$$
Radius $$\displaystyle=\sqrt { \frac { 49{ a }^{ 2 } }{ 4 } +\frac { { a }^{ 2 } }{ 4 } } =\frac { 5a }{ \sqrt { 2 } } $$
A helicopter is flying along the curve given by $$y - x^{3/2} =7, (x \ge 0)$$. A solider positioned at the point $$\left(\dfrac{1}{2}, 7\right)$$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is:
A tangent to the curve, $$y=f(x)$$ at $$P(x, y)$$ meets $$x-$$axis at $$A$$ and $$y-$$axis at $$B$$. If $$AP:BP=1:3$$ and $$f(1)=1$$, then the curve also passes through the point.
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Let $$f(x)=\dfrac{\sin \pi X}{x^2},x>0$$
Let $$x_1
If the normal to the curve $$y=f(x)$$ at the point $$(3,\,4)$$ make an angle $$\dfrac{3\pi}{4}$$ with the positive x-axis, then $$f'(3)$$ is
The points at which the tangents to the curve $$y=x^3-12x+18$$ are parallel to x-axis are?
Let $$y = f(x)$$ be a parabola, having its axis parallel to y-axis, which is touched by the line $$y =x$$ at $$x = 1$$, then
The volume of the greatest cylinder which can be inscribed in a cone of height $$30cm$$ and semi-vertical angle $$30^0$$ is
A rectangle is inscribed in an equilateral triangle of side length $$2a$$ units. The maximum area of this rectangle can be
Tangents are drawn to $$x^2+y^2=16$$ from the point $$P(0,h)$$. These tangents meet the x-axis at $$A$$ and $$B$$. If the area of $$\Delta PAB$$ is minimum, then $$h=?$$