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Standard XII

Mathematics

Directrix of Ellipse

If ax + by ...

Question

If ax $+$ by $+ c a = 0$ , where a,b,c>0 is the equation of the common radical axis of the coaxial system $(3 x^{2} + 3 y^{2} - 2 x - 2 y - 1) + λ (x^{2} + y^{2} - x - 2 y - 3) = 0$ , then $a b + b c +$ ca $=$

44

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Solution

The correct option is A $44$
Equation of radical axis is $s_{1} - s_{2} = 0$ .
Put $λ = 0$ . $s_{1} = x^{2} + y^{2} - \frac{2 x}{3} - \frac{2 y}{3} - \frac{1}{3} = 0$
and $s_{2} : x^{2} + λ^{2} - x - 2 y - 3 = 0$
$∴ s_{1} - s_{2} = 0$
$\frac{x}{3} + \frac{4 y}{3} + \frac{8}{3} = 0$
$x + 4 y + 8 = 0$
$a x + b y + c = 0$
$a = 1, b = 4$ & $c = 8$
$∴ a b + b c + a c = 4 + 32 + 8$
$= 44$

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If ax + by +ca=0, where a,b,c>0 is the equation of the common radical axis of the coaxial system (3x2+3y2−2x−2y−1)+λ(x2+y2−x−2y−3)=0 , then ab+bc+ ca =

The correct option is A 44Equation of radical axis is s1−s2=0.Put λ=0. s1=x2+y2−2x3−2y3−13=0and s2:x2+λ2−x−2y−3=0∴s1−s2=0x3+4y3+83=0x+4y+8=0ax+by+c=0a=1,b=4 & c=8∴ab+bc+ac=4+32+8=44

If ax $+$ by $+ c a = 0$ , where a,b,c>0 is the equation of the common radical axis of the coaxial system $(3 x^{2} + 3 y^{2} - 2 x - 2 y - 1) + λ (x^{2} + y^{2} - x - 2 y - 3) = 0$ , then $a b + b c +$ ca $=$