Conic Sections

Given, equation of circles are


and
Now, radical axis of circle and is




Radical axis of circle and is




For existence of radical centre

Given, circles are


Here,


Equation of common chord is



Since, and cut orthogonally




Then, from Eqs. (i), we get

Now, length of the common chord


Let = perpendicular distance from centre to the common chord


Now,

Given,



If

This is true only for point So, lies on the hyperbola

Given, line is

And equation of circle is
Here,
If given line touches the circle , then length of intercept





Alternative : If the given line is tangent to the circle, then the length of the perpendicular from the centre upon the line is equal to the radius of the circle.
ie,

Given equation is
Here,
As we know, that if tangents are drawn from an external points of a focal chord, then it will intersect on the directrix of the parabola.
Equation of directrix is
or
Which is the required equation.

Given,
Let be any point on the parabola

Also,



cannot be negative)


Points are or

Given that, parabola

Here, on comparing with

Let the parametric coordinate of parabola
is,
and the other coordinate of latusrectum is

Now, the equation tangent of parabola
...(i)
At

...(ii)



...(iii)
On solving Eqs. (i) and (ii)


___________________


...(iv)
From E (ii)



Hence, the intersection point of both tangent lying on .
ie, Which is the required locus.

Given that, equation of circle


Centre at
Radius
Also, the centre of another circle both circle touch externally. Then, distance between centres .

Now,

So, the radius of other circle is 4 .
Now, the equation of other circle touch externally to the circle is,

Let the radius of both circles are ' '.
Now, equation of circle with centre at is

and equation of circle with centre at is

Now, the equation common chord
Radical axis of and





Common chord

Given that, equation of parabola , let the parametric coordinate is .


Slope of normal
At
Now, the equation of normal to the parabola is


...(i)
Also, given the line
or ...(ii)
is normal to parabola, then
On comparing

Equation of conics

Equation of eccentricity of an ellipse




Focii of an ellipse

Given, focii of both conics are coincides.

Here is focii of second conic.]


Equation of eccentricity of second conic (hyperbola)

Equation of parabola Total length of focal chord and half length of focal chord Now, sum of the reciprocals of focal distances

Given,



Centre = (8, 0) and radius = 8
Since, the straight line touches the circle , therefore the length of perpendicular from the centre (8, 0) to the straight line is equal to the radius of the circle.
i.e.,




Taking positive sign, we get



Taking negative sign, we get

Any line through the point is


Now, radius of circle

If it is a tangent , then perpendicular from centre is equal to the above radius.










Putting the value of m = 2 in E (i) , we get


Again, putting the value of in E (i) , we get

The correct answer is D:
Equation of auxiliary circle of the ellipse,

Area of auxiliary circle
and area of an ellipse
Now, according to the question



Ecentricity (e)=
=
=
e=

Given, equation of parabola is . Here, Now, equation of tangent to the parabola in slope form is Also given that tangent to the parabola inclined at an angle of to the (+ve) direction of -axis. Then, [from E(i)]

Given hyperbola has focus which will lie on as it is focal chord.





Length of transverse axis

We have


and
Since,

The correct answer is (D) : II or IV.

Step 1: Understanding the Concept:
The problem relates to the geometric definition of conic sections obtained by intersecting a right circular double cone with a plane. The type of conic section depends on the relationship between the semi-vertical angle of the cone ( ) and the angle the intersecting plane makes with the vertical axis of the cone ( ).

Step 2: Key Formula or Approach:
Recall the standard classification of conic sections based on the angles and : - If , the intersecting plane is perpendicular to the axis, and the section is a circle. - If , the plane cuts entirely across one nappe of the cone, and the section is an ellipse. - If , the plane is parallel to a generator of the cone, and the section is a parabola. - If , the plane intersects both nappes of the double cone, resulting in a two-part curve called a hyperbola.

Step 3: Detailed Explanation:
Let's evaluate each statement based on the rules above. Statement I: "When , the section is hyperbola" This condition can be rewritten as . According to our classification, when the plane's angle with the axis is less than the semi-vertical angle, it cuts both halves of the cone, forming a hyperbola. Therefore, Statement I is true. Statement II: "When , the section is ellipse" When , the plane is exactly horizontal (perpendicular to the vertical axis). The resulting intersection is perfectly symmetric around the axis, forming a circle. While a circle is technically a special, degenerate case of an ellipse (where eccentricity ), in the context of standard conic section classification problems, "ellipse" refers to the general case where and the cross-section is elongated. A circle is treated as a distinct category. Thus, stating it is an ellipse when it is specifically a circle makes the statement false by convention in such multiple-choice questions.

Step 4: Final Answer:
Statement I is true and Statement II is false, which corresponds to option (1).