Question

C= 6 If P and R are two distinct points on the parabola y = cx? such that their tangent lines and normal lines form a rectangle PQRS that has a length that is twice as long as its width, then the area of that rectangle is a/b (reduced fraction). Find a +b.

Answer 1

y = 6x2 =) au to QC compare this with x²= 4my. Then 4m=6 :-) ㅗ ma 24 Then on will be any point the parabola Y=62² of the form (2mt, mt2) ) whene 24, Now dy d2 12% 12 x 2 mt 24 mt =t two مس ane titz t (2mt, mt?) Now fon fenen- any points Sabral (2miti, my?) (2 2m +2 , mtz?) the tangent at that points nespectively and Therefore to foom nectangle we have to two point on the parabola such that the points two adjacent side of the rectangle. z z tz Choose tangent at that ane

In that case tqtz = -1 Now, the tangent line equation (zont, mit 2) of the is at y amt 47 2- 2mty 2 =) y-mt, tix - zmly 2 tya - f 4 w similarly, the equation of the usbrunt line at (2mtz, mtz 2.) will be tza - = mtz Now we have find the point, bet tangent intersects. which this two 2 L = mti ta tn - y que Chammer's mile mt ㅖ m (12²-42) que 27 19 (4- tym ty t mty² m(txt2) mtitz (t2-to) (2-4) and at = mititz 3- my

The point .. where two tangent intenset is ( (tette),-) distance Now the we shall me nowne to froom (za 2mtu mtu 14?) and from (2mtz, (m Hits),-m -m) ) to (m (th++z),-- length (say, the first one) mtz One -them of other and is width (also given that tength. zu width). 2 L Length & width 2 2- >> [2m tymco in (tetta)] + m²(t+1) 2 + { [priz - Chan, ] + n°(t?)} >) (ty-tz)t (4+) 4 { (trte)' + (42–) *} 1 {**$) *(**)] -) (++++4207 -> (6*v) [b._sje qening that tra] [**304 =) +z = -) (14 72 ) - -> tz [] -) t = 21, +2:4

se pruebela we value t² 12, Can conside ong one of the Hy = 2 let's consider then t₂ = 2 Now, anea length x width Two points porrawala one Camti (2 mtz m 4 :) the m4?) (4m, 4m). mt22) (m point of intensection of the two (m (*+te), -m) (, -m) And tangents is nea (am - zien jent fm +m32 Vemum) (mm) X =m2 2.5 + 25 25 27 + 25 16 2 m 125 х 125 2 - m X 125 125 8x 242 8 125 4608 (given) - atb2 125+4608 = 47-33