Question

$\alpha ,\beta$ are the angles made by the tangents with the axis of the parabola drawn from the point $P$ to the paraboala $y^2=4ax$

List - I	List - II
A. $\cot \alpha \cot \beta =k$ , locus of P is	1. $kx=a$
B. $\tan \alpha +\tan \beta =k$ , locus of P is	2. $y=k(x-a)$
C. $\tan (\alpha +\beta )=k$ locus of P is	3. $kx=y$
D. $\tan \alpha \tan \beta =k$ , locus of P is	4. $xy=k$
	5. $x=ka$

The correct match for List - I from List - II

• A. $A-5,B-3,C-1,D-2$
  
• B. $A-3,B-2,C-1,D-5$
• C. $A-1,B-3,C-2,D-4$
• D. $A-4,B-2,C-1,D-5$

Answer 1

The correct option is A $A-5,B-3,C-1,D-2$

Equation of tangent is

$y=mx+\frac{a}{m}$

$m^{2}x-my+a=0$

$m_1+m_2=$ $\tan \alpha +\tan \beta =\frac{y}{x}$

$\tan \alpha \tan \beta =\frac{a}{x}$

a)$\cot \alpha \cot \beta =\frac{x}{a}=k\Rightarrow x=ka$

b) $\tan \alpha +\tan \beta =\frac{y}{x}=k\Rightarrow y=kx$

c) $\tan \alpha \times \tan \beta =\frac{a}{x}=k\Rightarrow x=\frac{a}{k}$

d) $\tan (\alpha +\beta )=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }=k=\frac{y}{x-a}$

$y=k(x-a)$