Parabola

The line and the circle intersect at the point and
Among the given options, only passes through both these points.

Given equation can be rewritten as


Here, focus is , directrix

the distance of the focus from the directrix


Latusrectum

Given




If line (1) touches parabola (2) then roots of quadratic equation (3) is equal

Given parabola is ....(1) Let and Slope of and slope of Since , ....(2) Let be the middle point of , then ....(3) and ....(4) From (4) , [ From (2) and (3)] Hence locus of is .

Given Parabola:

Given Circle:
Take a point on the parabola as
Solve equations simultaneously



We get the points as and .
The distance between and is given by,
Distance Formula
Distance
Focus of parabola has coordinates .
will form a right angled triangle with area square units.

Any tangent to parabola is
(i)
It touches the circle ,
if the length of perpendicular from the centre is equal to radius .





Hence, the required tangents are and .

Equation of parabola is . Its focus is at .

Answer (c) - t

The given equations are: Solving for the intersection points: Rearrange: Factoring: So, . For : Thus, length of the common chord: Now, the length of the latus rectum of the parabola is: Since and , we get:

The standard form of the given parabola is: The focus of the parabola is given by: Since the focus must lie between the lines and , we substitute the focus into the inequalities: Thus, the required range for is: