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Question

Statement -I : If – 2h = a + b, then one line of the pair of lines $a x to the power of 2 end exponent plus 2 h x y plus b y to the power of 2 end exponent equals 0$ bisects the angle between co-ordinate axes in positive quadrant.
Statement -II : If ax + y(2h + a) = 0 is a factor of $a x to the power of 2 end exponent plus 2 h x y plus b y to the power of 2 end exponent equals 0$ then b + 2h + a = 0.

Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
Statement-1 is True, Statement-2 is False
Statement-1 is False, Statement-2 is True

hint Hint:

From first statement , it implies that it lies in first quadrant and bisects the angle between coordinate axis, this means either x= y or x = -y

The correct answer is: Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

Statement -I : If – 2h = a + b, then one line of the pair of lines $a x to the power of 2 end exponent plus 2 h x y plus b y to the power of 2 end exponent equals 0$ bisects the angle between co-ordinate axes in positive quadrant.
From first statement , it implies that it lies in first quadrant and bisects the angle between coordinate axis, this means either x= y or x = -y
Check :
$rightwards double arrow – space 2 h space equals space a space plus space b rightwards double arrow 2 h space equals space minus left parenthesis a space plus space b right parenthesis S u b s t i t u t i n g space t h e space v a l u e space o f space 2 h space i n space a x squared space plus space 2 h x y space plus space b y squared space equals space space 0 comma space w e space g e t rightwards double arrow space a x squared space plus space minus left parenthesis a space plus space b right parenthesis x y space plus space b y squared space space equals space 0 rightwards double arrow space a x squared space space minus a x y space minus space b x y space plus space b y squared space space equals space 0 rightwards double arrow a x left parenthesis x space minus space y right parenthesis space minus b y left parenthesis x minus y right parenthesis space equals space 0 rightwards double arrow left parenthesis a x space minus b y right parenthesis left parenthesis x space minus space y right parenthesis space equals space 0 rightwards double arrow space e i t h e r space a x space equals space b y space o r space x space equals space y T h i s space p r o v e s space s t a t e m e n t space 1 space i s space c o r r e c t$

Statement -II : If ax + y(2h + a) = 0 is a factor of $a x to the power of 2 end exponent plus 2 h x y plus b y to the power of 2 end exponent equals 0$ then b + 2h + a = 0.
$b space equals space minus 2 h space minus space a F r o m space a b o v e space a x space minus b y space equals space 0 S u b s t i u t i n g space t h e space v a l u e space o f space b space i n space a b o v e a x space minus space left parenthesis negative 2 h space minus space a right parenthesis y space equals space 0 a x space plus 2 h y space plus space a y space equals space 0 a x space plus space y left parenthesis 2 h plus a right parenthesis space equals space 0 S t a t e m e n t space 2 space i s space c o r r e c t B u t space w e space d i d n o t space u s e space s t a t e m e n t space 2 space t o space p r o v e space s t a t e m e n t space 1$

Thus, Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.

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