Question

Two straight lines are perpendicular to each other.One of them touches the parabola $y^2=4a(x+a)$ and the other touches $y^2=4b(x+b)$ .The locus of the point of intersection of the two lines is

• A. x + a = 0
• B. x + b = 0
• C. x + a + b = 0
• D. x – a – b = 0

Answer 1

The correct option is C x + a + b = 0

Equation of tangent to the parabola $y^2=4a(x+a)isy=m(x+a)+\frac{a}{m}..........\left(1\right)$
Equation of tangent to the parabola $y^2=4b(x+b)isy=m_1(x+b)+\frac{b}{m_1}..........\left(2\right)$
Since the tangents are perpendicular $m{m}_1=-1\Rightarrow m_1=-\frac{1}{m}$
$\therefore$ (2) can be written as $y=\left(\frac{-1}{m}\right)(x+b)-bm.......\left(3\right)$
$\left(1\right)-\left(3\right)\Rightarrow 0=x(m+\frac{1}{m})+a(1+\frac{1}{m})+b(m+\frac{1}{m})\Rightarrow x+a+b=0$
Equation of the locus is x +a+b = 0