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Two tangents on a parabola are x-y=0 and x+y=0. S(2,3) is the focus of the parabola. If P and Q are ends of the focal chord of the parabola, then (1)/(SP)+(1)/(SQ)= |
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Answer» `2sqrt(13)//3`  We know that foot of perpendicular from the focus upon a tangent lies on the tangent at the VERTEX of the parabola. Now, if the foot of perpendicular of (2,3) on the line x-y=0 is `(x_(1),y_(1))` , then `(x_(1)-2)/(1)=(y_(1)-3)/(-1)=(-(2-3))/(2)` `orx_(1)=(5)/(2)andy_(1)=(5)/(2)` If the foot of perpendicular of (2,3) on the line x+y=0 is `(x_(2),y_(2))`, then `(x_(2)-2)/(1)=(y_(2)-3)/(1)=-(2+3)/(2)` `orx_(2)=-(1)/(2)andy_(2)=(1)/(2)` Now, the tangent at the vertex passes through the points `(5//2,5//2)and(-1//2,1//2)`. Then, its equation is `y-(1)/(2)=(2)/(3)(x+(1)/(2))` `or4x-6y+5=0` The length of latus rectum of the parabola is `4xx` (DISTANCE of locus from tangent at vertex) `=4xx|(8-18+5)/(sqrt(52))|=(10)/(sqrt(13))` Also, the distance between the focus and the tangent at vertex is `5//sqrt(13)` We know that `(1)/(SP)+(1)/(SQ)=(1)/(a)` where a is `1//4`th of the length of latus rectum. Therefore, `(1)/(SP)+(1)/(SQ)=(2sqrt(13))/(5)` |
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