You are given three lenses `L_1,L_2` and `L_3` each of focal lenth 15 cm.An object is kept at 20 cm in front of `L_1` as shown in Fig.2.87.

The real image is formed at the focus I of `L_3`.Find the separations between `L_1,L_2` and `L_3`.
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In the fig. l||m angle 1 and 2 are in the ratio 2:3. find all the angles

Which of the following lenses L_1 , L_2 and L_3 will you select as objective and eyepiece for constructing best possible (i) telescope and (ii) microscope? Give reason to support your answer.

A line L is a drawn from P(4,3) to meet the lines L_1a n dL_2 given by 3x+4y+5=0 and 3x+4y+15=0 at points A and B , respectively. From A , a line perpendicular to L is drawn meeting the line L_2 at A_1 Similarly, from point B ,a line perpendicular to L is drawn meeting the line L_1 at B_1 Thus, a parallelogram A A_1B B_1 is formed. Then the equation of L so that the area of the parallelogram A A_1B B_1 is formed. Then the equation of L so that the area of the parallelogram AA 1 ​ BB 1 ​ is least is

If (-2,6) is the image of the point (4,2) with respect to line L=0, then find the equation of line L.

For the following figures which one is the line l_1 or l_2 :

Consider three planes P_1 : x-y + z = 1 , P_2 : x + y-z=-1 and P_3 : x-3y + 3z = 2 Let L_1, L_2 and L_3 be the lines of intersection of the planes P_2 and P_3 , P_3 and P_1 and P_1 and P_2 respectively. Statement 1: At least two of the lines L_1, L_2 and L_3 are non-parallel . Statement 2:The three planes do not have a common point

Show that relation R = {(L_1, L_2) : L_1 is parallel to L_2 }, defined on the set of all lines, is an equivalence relation.

Show that the relation R defined in the set A of all lines as R = {(L_1, L_2): L_1 and L_2 are parallel lines} is an equivalence relation.

Let L_(1),L_(2),L_(3) be three straight lines a plane and n be the number of circles touching all the lines . Find the value of n.

Two planes P_1 and P_2 pass through origin. Two lines L_1 and L_2 also passing through origin are such that L_1 lies on P_1 but not on P_2 , L_2 lies on P_2 but not on P_1 . A,B, C are there points other than origin, then prove that the permutation [A', B', C'] of [A, B, C] exists. Such that: A lies on L_1 , B lies on P_1 not on L_1 , C does not lie on P_1 .