 The question says if the focus of the parabola y-k square is equal to 4 times x-h always lies between the lines x plus 1 is equal to 1 and x plus y equal to 3 Then which of the following options are correct Now friends when we start solving this problem One thing is for sure that we need to know the location of the focus of this parabola, right? So we need to know where does the focus of this parabola lie So under the roadmap, I would need to find the focus of the parabola second thing I would need to know is The relative position of this focus with respect to these two lines correct, so the second thing that we need to work on is the relative position of The focus With respect to the given lines Right, so with this roadmap in our mind will start solving this problem. All right, so When we talk about this parabola clearly, it's the case of a Shifted vertex Correct here It is not a case of standard parabola a standard parabola looks like y square is equal to 4x or y square is equal to minus 4ax or x square is equal to 4 a y or x square is equal to minus 4 a y in this case It is a case of shifted parabola now when you see this y minus k and h x minus h it should be clear in your mind that they are talking about the position of the vertex at h comma k Correct, so let us catch this first So, let's say this is my y-axis This is my x-axis okay, and We have a parabola of This nature I'm assuming my h and k both to be positive. So vertex is lying at h comma k Vertex is lying at h comma k Now we all know that This is called the axis of the parabola And focus is located at a distance of a from the axis of the parabola correct Now a here is one. This is your a correct So it's clear that your focus Let me call it as s We lie on or we lie at the point h plus 1 comma k Correct, so now the position of the focus is known to us Next is we are going to talk about the relative position of the focus with respect to the given lines So, let me do one more thing. Let me sketch these two lines quickly So let's say this is my y-axis. This is my x-axis Okay, and let's assume that this line is my line x plus y equal to 1 and this line is my line x plus y equal to 3 Okay, so let me name these lines This is my line x plus y equal to 1 or we can say x plus y minus 1 equal to 0 and This line is x plus y minus 3 equal to 0 Now our focus s lies somewhere in between these two lines Now for this we have to recall the concept of relative position of Two points with respect to a line. Now you would say where is the second point? We only have the focus s So we can take the second point as our origin Right, so origin could be taken as our second point Okay, so now we clearly can see that origin and s are located on Opposite sides of the line x plus y equal to 1 That means when you substitute x plus y minus 1 with the point zero zero Whatever number you should get the sign of that number should be opposite to what you should get when you substitute the point s Remember your s is h plus 1 comma k Right, so what I'm trying to say here is that when you substitute Origin in this right so put origin that is 0 comma 0 You realize you will get 0 plus 0 minus 1 which is a Negative 1 correct now when you substitute h plus 1 comma k So let's put h plus 1 comma k. I would get h plus 1 Plus k minus 1 which is nothing but h plus k Since they are on the opposite side Okay, since since origin and H plus 1 comma k lie on opposite sides of X plus y minus 1 equal to 0 that means Both should be of the opposite sign. So this and this must be of the opposite sign So if minus 1 is negative h plus 1 should be positive. Let's say this is our first conclusion In a similar way friends You would realize that Origin and s lie on the same side of the line x plus y minus 3 equal to 0 Okay, so we'll repeat the process once again for this line. So let's say this is line L 1 Okay, so line L 2 will repeat the same process x plus y minus 3 So first we'll put Origin when we put origin we get 0 plus 0 minus 3 which is negative and Then when we put h plus 1 comma k Will get h plus 1 Plus k minus 3 which is nothing but h plus k minus 2 now since since 00 and H plus 1 comma k lie on same side Same side of the line x plus y minus 3 equal to 0 it means if this is negative Then this must also be negative Which implies? H plus k minus 2 should be negative Which means h plus k should be less than 2 Okay, now let me call this as a second situation So this says h plus k should be greater than 0 This says h plus k should be less than 2 which clearly implies your option number a is correct Because in that case h plus k should lie between 0 and 2