Straight Lines

Let given line be Here, Condition for perpendicular line Condition of pair of lines

Condition for conjugate line is

Given equation of circle is The equation of tangent to the circle is Also, equation of parabola is Focus of parabola is Since, the line passes through focus .

Centres and radii of given circles are and ,

Let be the point at which the circle touch. Using internal ratio formula,

The perpendicular of the normal to the parabola at is

Suppose, it meets the parabola at . If be the vertex of the parabola, then the combined equation of and is a homogeneous equation of second degree. Since, and are at right angles, then Coefficient of Coefficient of

Let the equation of circle be where, centre The centre of given circle is Condition of two circles cut. Also, circle touch the line . The perpendicular distance from centre to the circle is equal to radius of the circle. Hence, the locus of is

Given equations of line are and Let, and be the slope of the line and Now, On solving Eqs. (i) and (iii), we get and Equation of lines will be and and third line is given The points of intersection of these lines are and Area of triangle

S.D = = , S.D =

Given points and are collinear. Slope of Slope of Given points become and . Equation of line passing through and is Now, the distance from to the above line is Equation of line passing through and is Since, above line is passing through . On putting in E (i), we get or Perpendicular distance from to the above line is