 Hello students, welcome to Centrum Academy YouTube channel. So today we have brought for you a very very good question a very very nice question on coordinate geometry topic So let's first read the question and then attempt to solve it So the question says from an external point P pair of tangent lines are drawn to the parabola y square is equal to 4x if theta 1 and theta 2 be the inclination of these tangents with the axis of x Such that theta 1 plus theta 2 is equal to pi by 4 Then which of the following options represents the locus of the point P? So essentially it's a question where the idea of pair of tangents is involved and we have also been provided with the concept of angle of inclinations So let us first quickly recap these two concepts for Eric Koenig The pair of tangents is given by a general equation t square is equal to s s 1 I'm assuming all my viewers are only aware of what T s and s 1 stands for Now we have also been given the angle of inclination of these two Pair of tangents or the two tangents which basically constitute the pair So we know that m 1 is equal to tan of theta 1 and m 2 is equal to tan of theta 2 Right, so now theta 1 and theta 2 are related to each other by the fact that theta 1 plus theta 2 is equal to pi by 4 Okay, so can I say one thing that if this is true then tan of theta 1 plus theta 2 would actually be tan of pi by 4 Which is 1 and this further expands as per the compound angle identity of tan theta 1 plus theta 2 as this tan theta 1 plus tan theta 2 divided by 1 minus tan theta 1 tan theta 2 right So in short what is given to you as the locus condition that m 1 plus m 2 Divided by 1 minus m 1 m 2 is equal to 1 Let's call this locus condition to be c because this will be of great use to us while finding the locus Alright, now let's move on to get the pair of tangents equation by using our General equation which I am showing you with this cloud So let us first assume the point let the point P be h comma k because without this We will not be able to write our t and s 1 So what is t when I'm assuming the point P to be h comma k? So t expression will be y y 1 in this case our y 1 is k minus 2 x plus x 1 in this case it is h So this is our expression for t. What is s s is y square minus 4x? What is s 1 s 1 is k square minus 4 h Now let us put this in our general form of the equation of the pair of tangents So t square is equal to s s 1 will become will become y k minus 2 times x plus h the whole square equal to s s this is our s s 1 s 1 will be k square minus 4 h Okay, now for a pair of straight lines Let's take a theory again over here So we know that for a pair of straight lines a x square plus 2 h x y plus b y square Plus 2 g x plus 2 f y plus c equal to 0 We know that if m 1 and m 2 are the slopes of the lines constituting this pair then m 1 plus m 2 is minus 2 h by b and And m 1 m 2 is equal to a by b I hope my viewers are already aware of this concept which is present in the pair of straight lines topic So we will be utilizing this idea to actually get the locus. How let's see So I have to first write down this equation that you see on your screen circle with a box in terms of our x square x y y square and of course the last three terms are Insignificant for me. So I will not be wasting time finding that out So let's find out the coefficient of x square from here. So coefficient of x square will be for Okay, what will be the coefficient of x y so please note that x y will be formed from only the left-hand side term over here Which I'm showing with a square basis in the base. So this will give you the coefficient of x y as minus 4k Right and what about coefficient of y square coefficient of y square will come from k square on the left side and On the right side you have negative of k square minus 4h Okay, rest of the terms we need not write because they will not contribute to our solving of the problem So now from here we end up getting the equation of the pair of tangents as 4x square minus 4k x y plus 4h y square plus some terms which are of no significance to us So m1 plus m2 from here would be negative 2h which is negative of negative 4k by b which in this case is 4h and this becomes k by h on simplification and m1 into m2 m1 into m2 would be a by b which is 4 by 4h which is nothing but 1 by h Now, let us use these in our condition number C Right, so I hope you all of you remember the condition number C which we have on the top here m1 plus m2 by 1 minus m1 m2 equal to 1. So let us use it there. So m1 plus m2 is So let me rewrite that down first Yeah, so in this condition we are going to put this so put this in this condition So we end up getting k by h divided by 1 minus 1 by h equal to 1 Which on simplification becomes k by h minus 1 equal to 1 Which means k is equal to h minus 1 or further you can write it as h minus k minus 1 equal to 0 Now time to generalize let us generalize here because we are looking for the locus So on generalization, we end up getting the equation to be x minus y minus 1 equal to 0 So this is my locus my dear friends. Let us see which option matches with this That's clearly option number C. That's clearly option number C So I hope you learned a lot of concepts to this simple problem solving Please do like subscribe and comment and press on the bell icon to get further notifications. Thank you so much