Question

If the line $3x+4y=12$ is a tangent to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=2$ then find the point of contact.

• A. (8, 6)
• B. (6, 8)
• C. (3, 4)
• D. (4, 8)

Answer 1

The correct option is A

(8, 6)

We will assume the point to be $P(x_1,y_1)$. We know how to find the tangent at $(x_1,y_1)$ to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=2$

It is $\frac{x{x}_1}{16}+\frac{y{y}_1}{9}=2$ (T=0)

We will make the R.H.S equal to 12 So that we can compare it with the given equation of tangent

$\Rightarrow 6\times \frac{x{x}_1}{16}+6\times \frac{y{y}_1}{9}=6\times 2=12$...(1)

$3x+4y=12$...(2)

is the equation given

(1)and (2) both represent the same equation

Comparing both the equations, we get

$\frac{6x_1}{16}=3and\frac{6y_1}{9}=4$

$\Rightarrow x_1=8and{y}_1=6$

$\Rightarrow (8,6)$ is the point at which the tangent is drawn.