A rod of length eight units moves such that its ends A and B always lie on the lines x−y+2=0 and y+2=0, respectively. If the locus of the point P, that divides the rod AB internally in the ratio 2:1 is 9(x2+αy2+βxy+γx+28y)−76=0, then α−β−γ is equal to:
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Step-by-Step Solution
Step 1: Define coordinates of A, B, and P
Let the coordinates of point A be (xA,yA) and point B be (xB,yB). Since A lies on the line x−y+2=0, we have yA=xA+2. Since B lies on the line y+2=0, we have yB=−2. Let P(x, y) be the point that divides AB in the ratio 2:1.
Step 2: Apply Section Formula
Using the section formula for internal division, we can express xA,yA,xB,yB in terms of x, y. From the equations, we get xA=3x−2xB and yA=3y−2yB.
Step 3: Substitute coordinates into line equations
Substitute yA=xA+2 and yB=−2 into the expressions from the section formula. This gives us 3y−2(−2)=(3x−2xB)+2, which simplifies to 3y+4=3x−2xB+2. Rearranging this, we find 2xB=3x−3y−2, so xB=23x−3y−2.
Step 4: Calculate xA and yA
Now we can find xA and yA using the expressions derived earlier. Substituting xB into the equation for xA, we get xA=3x−(3x−3y−2)=3y+2. Then, using yA=xA+2, we find yA=(3y+2)+2=3y+4.
Step 5: Use distance formula for AB
The length of the rod AB is 8 units. We use the distance formula to set up an equation: (xA−xB)2+(yA−yB)2=82. Substitute the expressions for xA,yA,xB,yB in terms of x and y into this equation.
Step 6: Substitute and simplify to find locus
Substitute the expressions for xA,yA,xB,yB into the distance formula. This gives us: ((3y+2)−23x−3y−2)2+((3y+4)−(−2))2=64 Simplify the terms inside the parentheses: (26y+4−3x+3y+2)2+(3y+6)2=64 (2−3x+9y+6)2+(3y+6)2=64 49(−x+3y+2)2+9(y+2)2=64 Multiply by 4: 9(−x+3y+2)2+36(y+2)2=256 Expand the squares: 9(x2+9y2+4−6xy−4x+12y)+36(y2+4y+4)=256 9x2+81y2+36−54xy−36x+108y+36y2+144y+144=256 Combine like terms: 9x2+117y2−54xy−36x+252y+180=256 9x2+117y2−54xy−36x+252y−76=0 Divide by 9: x2+13y2−6xy−4x+28y−976=0 Multiply by 9 again to match the given form: 9(x2+13y2−6xy−4x+28y)−76=0 Comparing this with the given locus 9(x2+αy2+βxy+γx+28y)−76=0, we find: α=13, β=−6, γ=−4.
Step 7: Calculate α−β−γ
Now we substitute the values of α=13, β=−6, and γ=−4 into the expression α−β−γ.