Question contains Statement-1 and Statement-2 and has the...

Question contains Statement-1 and Statement-2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct. (a) Statement-1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1 (b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1 (c) Statement- 1 is True, Statement- 2 is False (d) Statement- 1 is False, Statement- 2 is True Statement 1 Equation of a pair of lines is given by $3 x^{2}-2 y^{2}+5 x y+19 x-4 y+6=0$ Let the coordinate axes $\mathrm{Ox}$ and Oy be rotated through an angle $\frac{\pi}{4}$ in the positive sense and let the new system of axes be OX and OY. If the equation of the pair of lines in the new coordinate system is $\mathrm{AX}^{2}+2 \mathrm{HXY}+\mathrm{BY}^{2}+2 \mathrm{GX}+$ $2 \mathrm{FY}+\mathrm{C}=0$, then, $\mathrm{A}+\mathrm{B}=1$ and Statement 2 Let the equation of a pair of lines be given by $$a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$$ If the axes of coordinate are rotated through an angle $\theta$ in the positive sense without changing the origin and if the equation to the pair of lines in the new coordinate system is $$ \mathrm{AX}^{2}+2 \mathrm{HXY}+\mathrm{BY}^{2}+2 \mathrm{GX}+2 \mathrm{FY}+\mathrm{C}=0, \text { then, } \mathrm{a}+\mathrm{b}=\mathrm{A}+\mathrm{B} $$

Question contains Statement-1 and Statement-2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement-1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1
Equation of a pair of lines is given by $3 x^{2}-2 y^{2}+5 x y+19 x-4 y+6=0$ Let the coordinate axes $\mathrm{Ox}$ and Oy be rotated through an angle $\frac{\pi}{4}$ in the positive sense and let the new system of axes be OX and OY. If the equation of the pair of lines in the new coordinate system is $\mathrm{AX}^{2}+2 \mathrm{HXY}+\mathrm{BY}^{2}+2 \mathrm{GX}+$ $2 \mathrm{FY}+\mathrm{C}=0$, then, $\mathrm{A}+\mathrm{B}=1$
and
Statement 2
Let the equation of a pair of lines be given by
$$a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$$
If the axes of coordinate are rotated through an angle $\theta$ in the positive sense without changing the origin and if the equation to the pair of lines in the new coordinate system is
$$
\mathrm{AX}^{2}+2 \mathrm{HXY}+\mathrm{BY}^{2}+2 \mathrm{GX}+2 \mathrm{FY}+\mathrm{C}=0, \text { then, } \mathrm{a}+\mathrm{b}=\mathrm{A}+\mathrm{B}
$$

Key Concepts

Rotation of Axes

Rotation of Axes is a method used to change the coordinate system by turning it through a specified angle about the origin. This transformation simplifies equations with mixed terms (like xy) and enables easier analysis of geometric figures such as conic sections or pairs of lines. Importantly, when the origin remains fixed, certain algebraic combinations of coefficients, such as the sum of the squared term coefficients, are invariant under the rotation.

Equation of a Pair of Lines

An equation representing a pair of lines is a specific type of second-degree polynomial in two variables that factors into a product of two linear terms. This form includes both degenerate and non-degenerate cases and is fundamental in determining the intersection properties and angular relationships of the lines. Recognizing and manipulating such equations is essential in algebra and analytic geometry.

Invariance of Quadratic Form Coefficient Sums

For quadratic forms like those representing pairs of lines or conic sections, certain sum of coefficients (for example, the sum of the coefficients of x² and y²) remains unchanged when the coordinate system is rotated. This property results from the nature of rotational transformations that preserve distances and angles, and it forms a powerful tool for verifying consistency of equations before and after rotation.

Transcript

00:01 Problem we have been given that the question contains statement one and statement two and has following choices a b c and d out of which only one is correct now statement one is equation of a pair of lines is given by three x square minus two y square plus five x y plus 19 x minus four y plus six equal to zero let the coordinate axis o x and o y be rotated through an angle pi by four in the positive sense and let the new system of axis be o x and o y if the equation of the pair of lines in the new coordinate system says capital a capital x square plus two capital h capital x capital y plus capital b capital y square plus two g two capital g capital x plus two capital f capital y plus capital c equal to zero then capital a plus capital b equal to one and statement two uh i think i use the word capital many times the equation is nothing but all letters are in capital a x square plus 2 h xy plus b y square plus 2 g x plus 2 fy plus c equal to 0 and a plus b equal to 1 the second statement says let the equation of a pair of lines given by a x square plus 2 h x xy plus b y square plus 2 g x plus 2 fy plus c equal to 0 if the axis of coordinates are rotated through an angle theta in the positive sense without changing the origin and if the equation to the pair of lines in the new coordinate system says all letters in capitals a x square plus h x x xy plus b y c square plus 2g x plus 2 f5 plus c equal to 0 then small a plus small b equal to capital a plus capital b so this is the question let's proceed with the answer now consider statement 2 which i am abbreviating as st 2 now let x comma y denote the coordinates let small x comma y denote the coordinates of a point in the xy system and let capital x comma y denote the coordinates of the same point in the new system which is the capital x capital y system now small x would be equal to capital x cos theta minus capital y sine theta and small y would be equal to capital x sine theta plus capital y cos theta now substituting in the equation of a x square plus 2h xy plus b y square plus 2g x plus 2 f5 plus c equal to 0, the equation of the pair of lines in the new system would be...

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