Match the column

$\begin{matrix} Equation & Name of the curve 1) x^{2} - 2 x - y - 3 = 0 & P) Circle 2) x^{2} + 3 x y + 2 y^{2} - x - 4 y - 6 = 0 & Q) Parabola 3) x^{2} + y^{2} - 20 = 0 & R) Ellipse 4) 7 x^{2} + 7 y^{2} + 2 x y + 10 x - 10 y + 7 = 0 & S) Hyperbola 5) 6 x^{2} - x y - y^{2} - 23 x + 4 y + 15 = 0 & T) Pair of straight lines \end{matrix}$

1 - T, 2 - Q, 3 - P, 4 - R, 5 - S

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1 - Q, 2 - T, 3 - P, 4 - R, 5 - S

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1 - Q, 2 - T, 3 - P, 4 - S, 5 - R

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1 - Q, 2 - S, 3 - P, 4 - T, 5 - R

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Solution

The correct option is B
1 - Q, 2 - T, 3 - P, 4 - R, 5 - S

We have to decide the curve represented by the equations given. All of them are second degree curves. The systematic way of finding the conic is as follows.

Let the equation of the curve be

$a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ and

$△ = a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2}$

1) Find $△$

2) If $△ = 0$ , it represents a pair of straight lines

3) If $△ \neq 0$ , it's a conic

No we have to decide which conic for that we will compare $h^{2}$ and ab.

4) $h^{2} = a b \Rightarrow$ parabola

5) $h^{2} < a b \Rightarrow$ ellipse or circle

For circle, a = b and h = 0 otherwise it will be an ellipse

6) $h^{2} > a b \Rightarrow$ hyperbola

We will go through each equation and decide

P) $x^{2} - 2 x - y - 3 = 0$

$a = 1, b = 0, h = 0, g = - 1, f = - \frac{1}{2}, c = - 3$

$△ = 0 + 0 - 1 \times {(\frac{- 1}{2})}^{2} - 0 - 0 = \frac{- 1}{4}$

$△ \neq 0$

$h^{2} = 0$

ab = 0

$△$ = 0 and $h^{2} = a b \Rightarrow$ parabola

Q) $x^{2} + 3 x y + 2 y^{2} - x - 4 y - 6 = 0$

$a = 1, b = 2, b = \frac{3}{2}, g = \frac{- 1}{2}, f = - 2, c = - 6$

$△ = 1 \times 2 \times - 6 + 2 \times - 2 \times \frac{- 1}{2} \times \frac{3}{2} - 1 \times (- 2)^{2} - 2 \times {(\frac{- 1}{2})}^{2} - (- 6) \times \frac{9}{4}$

$△ = 0$

$△ = 0 \Rightarrow$ pair of straight lines

R) $x^{2} + y^{2} - 20 = 0$

If a=b and h=0 ,it represents a circle

Here $a = b = 1$ and $h = 0 \Rightarrow$ circle

S) $7 x^{2} + 7 y^{2} + 2 x y + 10 x - 10 y + 7 = 0$

$a = 7, b = 7, h = 1, g = 5, f = - 5, c = 7$

$△ = 7 \times 7 \times 7 + 2 \times - 5 \times 5 \times 1 - 7 \times (- 5)^{2} - 7 \times 5^{2} - 7 \times 1^{2}$

$△ \neq 0$

$h^{2} = 1, a b = 44$

$△ \neq 0, b^{2} < a b \Rightarrow E l l i p s e$

T) $6 x^{2} - x y - y^{2} - 23 x + 4 y + 15 = 0$

$a = 6, b = - 1, h = \frac{- 1}{2}, g = \frac{- 23}{2}, f = 2, c = 15$

$△ \neq 0$

$h^{2} = {(\frac{- 1}{2})}^{2} = \frac{1}{4}, a b = - 6$

$h^{2} > a b \Rightarrow H y p e r b o l a$

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Match the column EquationName of the curve1)x2−2x−y−3=0P) Circle2)x2+3xy+2y2−x−4y−6=0Q) Parabola3)x2+y2−20=0R) Ellipse4)7x2+7y2+2xy+10x−10y+7=0S) Hyperbola5)6x2−xy−y2−23x+4y+15=0T) Pair of straight lines