Question

The equation of tangent to the curve $y^2=a{x}^2+b$ at point $(2,3)$ is $y=4x-5$ , then the values of $a$ and $b$ are

• A. $3,-5$
• B. $6,-5$
• C. $6,15$
• D. $6,-15$

Answer 1

Answer: (D)

Given, $y^2=a{x}^2+b^2...(i)$
$\therefore 2y\frac{dy}{dx}=2ax$
$\Rightarrow \frac{dy}{dx}=\frac{ax}{y}$
$\therefore$ Slope at $(2,3)={\left(\frac{dy}{dx}\right)}_{(2,3)}=\frac{2a}{3}$
But slope of given tangent is $m=4$
$\therefore \frac{2a}{3}=4\Rightarrow a=6$
Since, point $(2,3)$ lies on the curve so, it satisfies the equation of the curve
$\therefore (3{)}^2=6(2{)}^2+b$
$\Rightarrow b=-15$