The tangent and normal to the ellipse 3x2 + 5y2 = 32 at the point P(2, 2) meet the x-axis at Q and R, respectively. Then the area (in sq. units) of the triangle PQR is :

Question

The tangent and normal to the ellipse 3x2 + 5y2 = 32 at the point P(2, 2) meet the x-axis at Q and R, respectively. Then the area (in sq. units) of the triangle PQR is : (A) (14/3) (B) (16/3) (C) (68/15) (D) (34/15)

Tardigrade Admin · Accepted Answer

3x2 + 5y2 =32 (dy/dx) Bigg|(2,2) = - (3/5) Tangent y-2 = - (3/5)(x-2) ⇒ Q((16/3) ,0) Normal: y-2 = (5/3) (x-2) ⇒ R ((4/5),0) Area is =(1/2)(QR)×2 = QR = (68/15)

Q. The tangent and normal to the ellipse $3 x^{2} + 5 y^{2} = 32$ at the point $P (2, 2)$ meet the x-axis at $Q$ and $R$ , respectively. Then the area (in sq. units) of the triangle $PQR$ is :

$\frac{14}{3}$

$\frac{16}{3}$

$\frac{68}{15}$

$\frac{34}{15}$

$3 x^{2} + 5 y^{2} = 32$
$\frac{d y}{d x} ∣ ∣_{(2, 2)} = - \frac{3}{5}$
Tangent $y - 2 = - \frac{3}{5} (x - 2) \Rightarrow Q (\frac{16}{3}, 0)$
Normal : $y - 2 = \frac{5}{3} (x - 2) \Rightarrow R (\frac{4}{5}, 0)$
Area is $= \frac{1}{2} (QR) \times 2 = QR = \frac{68}{15}$

Q. The tangent and normal to the ellipse 3x2+5y2=32 at the point P(2,2) meet the x-axis at Q and R, respectively. Then the area (in sq. units) of the triangle PQR is :

314​

316​

1568​

1534​